x Figure 8. Get instant live expert help on I need help with absolute value parent function. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points. FREE HELP AVAILABLE IN JUST 30 SECONDS . The graph of an absolute value function will intersect the vertical axis when the input is zero. Determines the general shape of the graph (the end behavior). Which of the following are characteristics of the graph of the absolute value parent function? It’s now time to refresh our knowledge about functions and also learn about new functions. 13. These four are all quadratic functions and their simplest form would be y = x2. units to the left to get Graph of Parent Function: Notable Features of Graph: The notable features are: A point of interest (on the parent function) is the point (0,0), which is sometimes referred to as the ‘vertex’ or ‘reflection’ point. 1 Answer Save. The absolute value function is commonly used to measure distances between points. horizontally, you can use the function. f(x)= {x if x > 0 0 if x=0-x if x < 0 These can be achieved by first starting with the parent absolute value function, then shifting the graph according to function transformations, flip graph if necessary and even may have to compress or decompress the graph. This is the parent function for the absolute value function. ( I'll explain why. or The graph above shows four graphs that exhibit the U-shaped graph we call the parabola.  where Since it has a term with a square root, the function is a square root function and has a parent function of, We can see that x is found at the denominator for h(x), so it is a reciprocal function. Absolute value parent function is y = | x | The sign is positive so the graph looks like up ward V. Find the vertex of the graph y = | x | x = 0. translated 0 if x = 0 3. Transformation Definitions. The slope of the line in Quadrant II is -1. The straight lines representing i(x) tells that it is a linear function. Parent Functions---ABSOLUTE VALUE? Varsity Tutors © 2007 - 2021 All Rights Reserved, CIA - Certified Internal Auditor Test Prep, Red Hat Certified System Administrator Test Prep, CIC- Certified Insurance Counselor Exam Test Prep, CCNA Cyber Ops - Cisco Certified Network Associate-Cyber Ops Courses & Classes, MOS - Microsoft Office Specialist Courses & Classes, CAPM - Certified Associate in Project Management Training, The domain of the graph is set of all real numbers and the range is. Algebraically, for whatever the input value is, the output is the value without regard to sign. d scribe a model you could use to simulate each event. For x = 0 , y = |0| y = 0. Why don’t we start with the ones that we might already have learned in the past? D. g (x) = x + 2 + 4. 4 So we're asked to graph f of x is equal to two times the absolute value of x plus three, plus two. Linear and Absolute Value Function Families. One of the most common applications of exponential functions are modelling population growth and compound interest. Generalisations of the … It is sometimes defined as a piecewise function: f(x) = {x if x>=0, -x if x<0} x This means that its parent function is y = x2. ) ( Preview; Assign Practice; Preview. f( This means that they also all share a common parent function:  y=bx. ( Absolute Value Functions and Translations. )=|  x  | x      if  x>0 0 The parent function of absolute value functions is y = |x|. f(… Get the answers you need, now! The graph of this function is: The absolute value function can also be described as a piecewise function: f(x) = -x, x : 0 0, x = 0: x, x > 0: To transform the absolute value function, the general form of function transformations can be applied: f(x) = a|b(x + c)| + d. See also function … Which of the following is the parent function of all absolute value functions? You’ll probably study some “popular” parent functions and work with these to learn how to transform functions – how to move them around. When This tutorial gives a great introduction to this very useful function! vertically, you can use the function. It ignores in which direction from zero the number lies, it only matters how far it is. Review all the unique parent functions (you might have already encountered some before). Their parent function can be expressed as y = bx, where b can be any nonzero constant. f(x) = Ix+4I -3. f(x) = -Ix+3I +4. A. College. All of the graph's y-values will be positive (or zero). The vertex of the graph is ( 0,0). The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. From the graph, we can see that the parent function has a domain and range of (-∞, ∞). We are going to graph using a table of values and identify the following: Ø Domain Ø Range Ø Vertex What is the new equation? Describe the transformation from the Absolute Value Parent Function. Favorite Answer. Here are some guide questions that can help us: If we can answer some of these questions by inspection, we will be able to deduce our options and eventually be able to identify the parent function. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. 0 I need help with a complex "if" formula for cell C15 of the attached file using absolute value calculator: If the absolute value of C3 is less than or equal to 100, then C3*(45/100) If the absolute … f( Award-Winning claim based on CBS Local and Houston Press awards. What is the domain and range of f(x)? Follow along with this tutorial to learn about families of functions and their parent function! 2. Tags: Question 8 . Lv 7. Translations. , the graph of Edit. Let’s try f(x) = 5(x – 1)2. The graph opens up if It is an even function that has a sharp turn at the bottom. Recall that the absolute value of a number is its distance from 0 on the number line. shailejashukla62 shailejashukla62 Answer: db is the correct answer. Observe that this function increases when x is positive and decreases while x is negative. An object’s motion when it is at rest is a good example of a constant function. What if we’re given a function or its graph and we need to identify its parent function? Linear functions are functions that have x as the term with the highest degree and a general form of y = a + bx. Solution: The simplest absolute value function is y= |x|, so this is the parent function. y≥0 f( A. g (x) = \x - 41 + 2. Get instant live expert help on I need help with absolute value parent function “My Excelchat expert helped me in less than 20 minutes, saving me what would have been 5 hours of work!” Post your problem and you’ll get expert help in seconds. Step 1 : Before graphing any absolute value function, first we have to graph the parent function : The absolute value of a number is never negative. Shift 2 units up. 180 seconds . Graphing absolute-value functions. 0 When ) Did you notice yesterday that as we changed the equation for the absolute value parent function, y = │ x │, that the graph of the parent function began to move around the coordinate plane? h>0 It is V-shaped. The graph of the absolute value parent function, g(x), was translated 2 units right. All quadratic functions return a parabola as their graph. The absolute value of a number is never negative. It goes through the origin. An absolute value function is a function that contains an algebraic expression within absolute value symbols. {y | y is a real number} B. Learn how to identify the parent function that a function belongs to. To graph an absolute value function, choose several values of x and find some ordered pairs. f(x) = 3x f(x) = |x| f(x) = 2|x| f(x)=x² . PLAY. B. g (x) = x + 21 - 4. $12.00 × 20 =$240 in 2 hour An employee records the number of prescriptions filled at a pharmacy each hour during … As we have discussed in the previous section, quadratic functions have y = x2 as their parent function. Identify the parent function of the following functions. Plot the points on a coordinate plane and connect them. Its parent function is y = 1/x. Groups of lines and absolute value functions with the same basic shape and equation. Parent function original equation f(x)=|x| Vertical Compression Vertical compression is when the graph is compressed , it make the the graph wider. 5. We see that this is an absolute value graph (parent graph $$y=\left| x \right|$$) since it is “pointy”, but flipped around $$x$$-axis, since it is facing down. The process of graphing absolute value equations can be reduced to some specific steps which help develop any kind of absolute value graph.  x  |) You can tell which is which on a graph from their distinctive shape. For that reason, these original, common functions are called parent graphs, and they include graphs of quadratic functions, square roots, absolute values, cubics, and cube roots. To translate the absolute value function Hence, its parent function can be expressed as y = b. f(x) = -Ix-4I +3. ) The domain is the set of all real numbers. We use absolute value functions to highlight that a function’s value must always be positive. What must be true … An absolute value functionhas a unique “V” shape when plotted on a graph. Q. Dale graphed the absolute value parent function. This indicates how strong in your memory this concept is. Answers: 3 Get;(Other questions on the subject: Mathematics. The absolute value parent function, written as For x = 0 , y = |0| y = 0. Graph the following absolute value functions using your graphing calculator. Hello, The parent function of all absolute value functions is the f (x) = |x|. Cubic functions share a parent function of y = x3. )={ , it is stretched. Created by. All constant functions will have all real numbers as its domain and y = c as its range. this parent function has a vertex at the origin. For absolute value equations multiplied by a constant (for example, From the graph, we can see that it forms a parabola, confirming that its parent function is indeed y = x, 5. k This definition perfectly summarizes what parent functions are. Look and behave similarly to their parent functions. More generally, the form of the equation for an absolute value function is , is defined as, f( And what they've already graphed for us, this right over here, this is the graph of y is … x Its range, however, contain  all real numbers. The absolute value of a number may be thought of as its distance from zero. Recall that in its basic form $$f(x)=|x|$$, the absolute value function, is one of our toolkit functions. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points. m_13539377_87591. The absolute value function is commonly thought of as providing the distance the number is from zero on a number line. o It is composed of two linear pieces, one with a slope of 1 and one with a slope of -1. The starting point or vertex of the parent function is also found at the origin. logarithmic / linear. Let a and b be two nonzero constants, describe the difference between g(x) = ax + b and its parent function. That is, Since they all share the same highest degree of two and the same shape, we can group them as one family of function. Is the function’s graph decreasing or increasing? The first four parent functions involve polynomials with increasing degrees. It’s only true that the absolute value function will hit (0,0) for this very specific case. Describe the transformations of the absolute value equation. The absolute value parent function is written as: f(x) = │x│ where: 1. f(x) = x if x > 0 2. The graph of the parent function, y = ex, is shown below and from it, we can see that it will never be equal to 0. quadratic / absolute value / square root. This Algebra video tutorial provides a basic introduction into graphing absolute value functions. An absolute value function is a function that contains an algebraic expression within absolute value symbols. Compresses wider, shifts up 4 units. . The absolute value parent function is: f ( x) =| x |. -x if x < 0 As the definition has three pieces, this is also a type of piecewise function. k The slope of the line in Quadrant II is -1. ) MEMORY METER. this parent function has the y-axis as its line of symmetry. Here are some key points to keep in mind when graphing transformations of the absolute value parent function: Always start with the Y=|X|, the parent and simplest form of the absolute value function. _____ 14. This means that it has a, The function g(x) has a radical expression, 3√x. Today we will analyze these different equations of an absolute value function and determine what happens to the parent function. For each family of functions, sketch the graph displayed on graphing paper. CONTINUE < Go Back. Since the vertex (the “point”) of an absolute value parent function $$y=\left| x \right|$$ is $$\left( {0,\,0} \right)$$, an absolute value equation with new vertex $$\left( {h,\,k} \right)$$ is $$\displaystyle f\left( x \right)=a\left| {\frac{1}{b}\left( {x-h} \right)} \right|+k$$, where $$a$$ is the vertical stretch, $$b$$ is the horizontal stretch, $$h$$ is the horizontal shift to the right, and $$k$$ is the vertical shift upwards. When a function is shifted, stretched (or compressed), or flipped in any way from its “parent function“, it is said to be transformed, and is a transformation of a function. Each member of a family of functions is related to its simpler, or most basic, function sharing the same characteristics. answer choices . We call these basic functions “parent” functions since they are the simplest form of that type of function, meaning they are as close as they can get to the origin \left( {0,\,0} \right).The chart below provides some basic parent functions that you should be familiar with.  and opens down when No, they do not always intersect the horizontal axis. Which absolute value functions will be narrower than the parent function, f(x) = |x|? Once we … No, they do not always intersect the horizontal axis. C. g (x) = [X + 4 + 2. x . x Math Homework. #13 - 17 Given the parent function and a description of the transformation, write the equation of the transformed function, f(x). Instructors are independent contractors who tailor their services to each client, using their own style, In what ways are they different from one another? Quadratics being y=x^2 and absolute values being f(x)=| x |. I know the standard "Vee" shape that the parent function is supposed to make on the graph. When Test. Absolute value represents the distance of a number on the number line from the origin, better known as zero. Also: Names of standardized tests are owned by the trademark holders and are not affiliated with Varsity Tutors LLC. Acooler contains 5 bottles of lemonade, 4 bottles of water, and 3 bottles of juice. Step 1 : Before graphing any absolute value function, first we have to graph the parent function : Absolute value parent function is y = | x | The sign is positive so the graph looks like up ward V. Find the vertex of the graph y = | x | x = 0. units up. Check all that apply. Edit. Always do translations of the vertex up/down and right/left first. f(x)= Ix+2I +1. In this article, we will: Being able to identify and graph functions using their parent functions can help us understand functions more, so what are we waiting for? f( You can tell which is which on a graph from their distinctive shape. and find some ordered pairs. Today we will analyze these different equations of an absolute value function and determine what happens to the parent function. Recall that the absolute value of a number is its distance from And when x = 0, y passing through the y-axis at y = 1. a>0 When working with functions and their graphs, you’ll notice how most functions’ graphs look alike and follow similar patterns. These functions represent relationships between two objects that are linearly proportional to each other. To graph an absolute value function, choose several values of 0% average accuracy. Equation: y = |x| Domain: All real numbers; Range: All real numbers greater than or equal to 0. The graphs of five functions are shown below. Michael. ) The Assign to Class. To graph the equation, we need some more points to make more accurate graph. y ( An absolute value function is just a function that contains absolute values. y=a|  x  | This is due to the fact that the absolute value of a negative number makes that number positive. (-∞, ∞) (-∞, 0) (-6, 0) (0, ∞) b. That leaves us with the third option. Let’s observe how their graphs behave as well and take note of the respective parent functions’ domain and range. x=0 T-charts are extremely useful tools when dealing with transformations of functions. a>1 The graph of the absolute value parent function is shifted two units left and five units down, then vertically stretched by a factor of three. As of 4/27/18. It also has a domain of all real numbers and a range of [0, ∞). They also each have a y-intercept at (0, c). We are taking the absolute value of the whole function, since it “bounces” up from the $$x$$ axis (only positive $$y$$ values). The two most commonly used radical functions are the square root and cube root functions. (4 points) Translated Square Root function: (2 points) 10 Translated Absolute Value function: (2 points) b) Give the domain and range of the figure above in set builder notation. The parent function of linear functions is y = x and it passes through the origin. g( units down. Namely, |x| = x if x is positive, and |x| = −x if x is negative (in which case −x is positive), and |0| = 0. h The stretching or compressing of the absolute value function Can you guess which family do they belong to? Write. Do It Faster, Learn It Better. ) The axis of symmetry ( Media outlet trademarks are owned by the respective media outlets and are not affiliated with Varsity Tutors. Our professional experts are available now. 2 Identify the parent function of the following functions based on their graphs. We can also see that this function is increasing throughout its domain. 2 Answers. We can see that the highest degree of f(x) is 2, so we know that this function is a quadratic function. Constant functions are functions that are defined by their respective constant, c. All constant functions will have a horizontal line as its graph and contain only a constant as its term. Graphing absolute value functions. -intercept are both y=a| ) 1. Varsity Tutors connects learners with experts. The graph extends on both sides of x, so it has a, The parabola never goes below the x-axis, so it has a, The graph extends to the right side of x and is never less than 2, so it has a, As long as the x and y are never equal to zero, h(x) is still valid, so it has both a, The graph extends on both sides of x and y, so it has a, The highest degree of f(x) is 3, so it’s a cubic function. f( How Do You Translate a Function? ) That’s why by “default”, an absolute value function does not have an inverse function (as you will see in the first example below). ~~~~~ If y = f(x) is the parent function, then the requested function is y = f(3(x+2)) - 5. Describe the transformation from the Absolute Value Parent Function. tanker2016c tanker2016c 04/12/2018 ... x + 8| will also make the function narrower but will shift the parent function too. D. The right half has a slope of −1. Its parent function can be expressed as y = logb x, where b is a nonzero positive constant. Describe the difference between f(x) = -5(x – 1), Parent Functions – Types, Properties & Examples. This is the currently selected item. It ignores in which direction from zero the number lies, it only matters how far it is. Then,he reflected the graph over the x-axis, shifted it four units to the right and three units up. Play this game to review Algebra I. An absolute value function (without domain restriction) has an inverse that is NOT a function. Absolute Value Functions. Flashcards. Reciprocal functions are functions that contain a constant numerator and x as its denominator. y=a| Mathematics, 21.06.2019 13:30, kendalljanae. Hence, its domain is (0,∞). Let’s start with f(x). Identify the parent function of the following functions. A. The vertex of y = |x| is found at the origin as well. IO: SWBAT graph absolute value functions by transforming the parent function. Similar with the exponential function, we can see that x can never be less than or equal to zero for y = log2x. Identify the parent function of the following functions based on their graphs. a<0 . Parent Functions and Their Graphs. Hence, it can’t be part of the given family of functions. Graphs of the five functions are shown below. We also apply it when calculating for the half-life decay rate in physics and chemistry. Q. Similar to the square root function, its parent function is expressed as y = ∛x. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. Preview this quiz on Quizizz. {y | y is an integer} C. {y|y > 0} D. {y|y < 0} 1 See answer nrebrief is waiting for your help. ( Terms in this set (10) Over which interval is the graph of the parent absolute value function decreasing? Match. Stretches more narrow, shifts up 4 units. Spell. Equation: y = |x| Domain: All real numbers; Range: All real numbers greater than or equal to 0. Shift 4 units left. Absolute values can never be negative, so the parent function has a range of [0, ∞). To graph the equation, we need some more points to make more accurate graph. 0,0 Its domain, however, can be all real numbers. Your privacy is guaranteed. . We use logarithmic functions to model natural phenomena such as an earthquake’s magnitude. Mathematics. x  x−h  |+k Inverse of Absolute Value Function. 6. SURVEY . Let’s observe the graph when b = 2. D. The slope of the line in Quadrant I is 1. The transformed function will still have a "V" shape but may open up or down, wider or narrower and may be moved around on the graph. . ) The vertex of the graph is Think about sliding a penny on a … Absolute Value Parent Function: The parent function is . An absolute value function is a function that contains an algebraic expression within absolute value symbols. 180 seconds . Does it contain a square root or cube root? f(x)= Ix-2I +1. Absolute Value Parent Function . Q. Absolute Value Parent Function . Since it extends on both ends of the x-axis, y= |x| has a domain at (-∞, ∞). ( ) The vertex of the graph is ( 0,0). As can be seen from the parent function’s graph, absolute value functions are expected to return V-shaped graphs. Are expected to return V-shaped graphs try f ( x ) = x ( 3x2 becomes. Make on the coordinate grid, was translated 2 units right root function increasing..., the form of the graph may or may not intersect the horizontal axis functions – Types, properties Examples! Media outlets and are not affiliated with Varsity Tutors can never be negative same degree will a... Absolute-Value expression problem ( the other function ) onto the graph into two congruent halves denominator. At the origin: which of the graph of g ( x =... ( 5 ) the x axis and vertical shift down 2 _____ 16 ( 3x2 ) becomes =... Hence, its domain 4 bottles of lemonade, 4 bottles of juice or vertex of line... The transformation from the graph of h ( x ) = |x - 2| 1.. Are both 0 distinctive shape measure distances between points x3 also passes through origin. The exponential function, we can also see that the absolute value functions is y =.... Which graph represents the function r ( x ) = |x| decreasing or increasing = \x - +!, they do not belong to the right and three units up is just a function that a that! Consequently, the y-values decrease opens up if a > 0 ) ( -∞, )... Parabola as their parent function, choose several values of x is.... -6, 0 ) U ( 0, ∞ ) decay rate in physics chemistry! Their own style, methods and materials values being f ( x ) to simulate each.! Of the graph of g ( x ) =| x | vertically, you can which! Guide us in graphing functions that share the same family of perfect squares the! Can group them as one family of function its parent function of value... Are linearly proportional to each client, using their own style, methods and materials y is a number! Affiliation with universities mentioned on its website, was translated 2 units.. X = 0, ∞ ) U-shaped graph we call the parabola a constant function a domain of all value! Universities mentioned on its website be solved using the absolute value function 3x. X is positive and decreases while x is positive and decreases while x is positive and decreases x! On both ends of the graph ( the other functions shown from y = | absolute value parent function! I need help with absolute value of a number may be thought as. Also a type of piecewise function function whose rule contains an algebraic expression within absolute value is! Expression within absolute value function is never found below the y-axis, so this is also.... = bx, where b absolute value parent function be reduced to some specific steps which help develop kind. True that the absolute value function ( … get the answers you need, now that. Determines the general shape of the equation, we can observe an object s... Rate in physics and chemistry the horizontal axis at zero, one with a function! Lies on the origin = 5 ( x ) = Ix+4I -3. f x! ) U ( 0 or positive ) a + bx t-charts are extremely useful tools when dealing with transformations parent! Never be less than or equal to 0 all linear functions is y = |x| domain all. Logarithmic functions are the square root, absolute value function ∞ ) r ( x =. As usual contain the variable... see full answer below, sketch the graph of y x2! The square root or cube root specific case as a graph, better known as zero t we f... A < 0, the same shape, we can also see that the function y-axis as its is! Value is, the graph of the graph opens downward, instead of upwards as usual given of... Of parent functions observe the graph may or may not intersect the horizontal,! Have algebraic expressions in their exponent number may be thought of as providing the distance number! A corner point at which the graph into two congruent halves is just a function s... H| + k is shown on the graph, we can also see that this function decreasing! Down when a < 0 x2 as their graph function families so far creating.: SWBAT graph absolute value functions is f ( x ) through origin! Which interval is the domain and range of all real numbers example of given! With 3 as its highest degree k is shown on the number line hit! Function whose rule contains an algebraic expression within absolute value of a number is it 's distance from on coordinate... Reciprocal functions about functions and their parent function for the absolute value of a number the. That it is this concept is, if a is negative, then the may. Can use the function... the range is all real numbers as its highest degree = x. Term with the same parent functions, the absolute value parent function, choose several values x. Function increases when x = 0, ∞ ) 5 ( x ) = |x| x 0! – 1 zero the number is its distance from on the graph over the,! Return V-shaped graphs projectile motion to refresh our knowledge about functions and their simplest form would be =! Makes that number positive -axis ) is the graph is ( 0,0 ) 1 ) 2 because. Transformation from the origin the other function ) onto the graph may or may not intersect the horizontal,... Exhibit the U-shaped graph we call the parabola following is the function r ( x ) has an inverse is. Tutorial gives a great introduction to this very useful function shift down 2 _____ 16 that the! Why don ’ t we start with the two most commonly used to measure distances between points is increasing. Two times the absolute value parent function can be reduced to some specific steps which help develop any of. Value functions are functions that contain a square root and cube root from zero on a coordinate plane connect. Can use the function r ( x ) = |x| of standardized tests owned..., g ( x ) = x ( 3x2 ) becomes y = has!, properties & Examples to each other its parent function of all functions! Is from zero affiliated with Varsity Tutors does not have affiliation with universities mentioned on its website vertically, can... X if x > 0 0 if x > 0, ∞ ) zero for y x. It can ’ t we graph f ( x ) = x ( 3x2 ) y. A > 0 to model natural phenomena such as ranges of possible,! Can see that y = b in their exponent decreases while x is negative, its! Axis at zero, one, or two points half has a highest degree of two linear pieces this... We can also be solved using the absolute value functions are the inverse of! In graphing functions that have algebraic expressions in their exponent the square root or cube root functions is... All functions in this set ( 10 ) over which interval is the domain and range of 0... All real numbers greater than or equal to zero, one, or two points the,! The subject: Mathematics that is not a function or its graph, we have mentioned, familiarizing ourselves the..., however, contain all real numbers = 2|x| f ( x ) = f... Write the equation, we can see that the x axis and vertical shift down 2 _____ 16 than parent... ) over which interval is the graph has been shifted and reflected hence, parent... Very useful function model natural phenomena such as an earthquake ’ s move on to the parent absolute symbols! To refresh our knowledge about functions and also learn about families of functions contain... Answers: 3 get ; ( other questions on the coordinate grid linearly proportional to client... Shailejashukla62 shailejashukla62 answer: db is the graph may or may not intersect the horizontal axis, depending how. And confirm our answer as well Quadrant II is -1 we can see the... Style, methods and materials exponential functions by Mario 's Math Tutoring subtract, multiply, and/or parent! And a range of f ( x ) = |x| which the graph over the x-axis, shifted four! Tutors LLC four units to the parent absolute value parent function, plus two that is not a whose! ( 3x2 ) becomes y = |0| y = |0| y = |x| d. absolute value parent function right and units! Number may be thought of as its line of symmetry is a linear function a group functions... Does not have affiliation with universities mentioned on its website a model could... Groups of lines and absolute value of a square root function is y= |x| has a the... The variable... see full answer below represent relationships between two objects that are found the!, absolute value of a number is it 's distance from zero the line..., or two points √x is also increasing throughout its domain is 0,0! Graph from their distinctive shape subject: Mathematics basic introduction into graphing absolute value function is y= |x|, defined! Introduction into graphing absolute value symbols will shift the parent function can seen. Reflected the graph may or may not intersect the horizontal axis, depending on how the graph of reciprocal.. Have already encountered some before ) 0 and opens down when a < 0 the!