Notes for section 8.4: Transforming Quadratic Functions A ‘family of functions’ is a group of functions that have basic characteristics in common. For example: Linear functions from a family because they all have x to the first power and they all graph a line. Quadratic functions also form a family because Like all families, each quadratic function is different; they can be wider or narrower, or move up or down, left or right. A ‘parent function’ is the most basic function in a family. In linear functions, the parent function is f(x) = x or y = x In quadratic functions, the parent function is _________________________________ The graph of the parent quadratic function looks like: The axis of symmetry is _________________________________ The vertex is _____________________________________________ a = ______ b = _______ c = _________ Since a is __________________________________________________ The graphs of all functions are transformations of the parent function. A transformation is a change in position or size. *Remember that a determines ______________________________________________________________________________ a also determines whether the parabola will be wider or narrower than the parent function. If the value of a (forget the sign) is larger than 1, then the parabola will be narrower. *As the value of a gets bigger, ____________________________________________________ If the value of a (forget the sign) is less than 1 (a fraction or decimal), then the parabola will be wider. *As the value of a gets smaller, ___________________________________________________ 1
f(x) = x2
f(x) = 3x2
f(x) = x2 4
1
a = 1 a = 3 a = 4
b = 0 b = 0 b = 0 c = 0 c = 0 c = 0 *Remember the standard form of a quadratic function: y = ax2 + bx + c We will not address b in this section, but let’s talk about the effect c has on the parent function. You already know that c _____________________________________________________________________________________ c also shifts the parabola up or down. If c > 0, then the parabola shifts UP c units. If c < 0, then the parabola shifts DOWN c units. f(x) = x2
f(x) = x2 + 4
f(x) = x2 – 3 a = 1 a = 1 a = 1 b = 0 b = 0 b = 0 c = 0 c = 4 c = –3 Vertex at (0,0) vertex shifts __________________________ vertex shifts _________________________ Comparing graphs of quadratic functions to the parent function, f(x) = x2 Directions: Compare the graph of each function with the graph of f(x) = x2. !
𝑔 𝑥 = − 𝑥 + 2 !
a = _______ so the parabola will open ______________ and will be ___________________ than the parent function f(x) = x2 c = _______ so the vertex will shift _____________________________ and will be at ____________ ℎ 𝑥 = −𝑥 ! − 4 a = ________ so the parabola will open ______________ and will be ___________________ than the parent function f(x) = x2 c = _______ so the vertex will shift _____________________________ and will be at ____________ 𝑘 𝑥 = 2𝑥 ! − 3 a = _______ so the parabola will open ______________ and will be ___________________ than the parent function f(x) = x2 c = _______ so the vertex will shift _____________________________ and will be at ____________