Definition: The graph of a quadratic function is a U-shaped curve called a parabola. The axis of symmetry is a line that divides the parabola into two symmetrical halves. The vertex lies on the axis of symmetry and is the highest or lowest point on a parabola, where it changes direction. The parent graph of all quadratic functions is y = x².
General form: f(x) = ax² + bx + c
Axis of symmetry: x = -b/2a
Vertex: (-b/2a, c – b²/4a)
a > 0, graph opens up
a < 0, graph opens down
y intercept: (0, c)
Vertex form: f(x) = a(x-h)² + k,
Axis of symmetry: x = h
Vertex: (h, k)
k: indicates vertical shift (k < 0 moves down, k > 0 moves up)
h: indicates horizontal shift (h < 0 moves left, h > 0 moves right)
a: indicates vertical stretch, compression or reflection (a > 1 for vertical stretch, 0 < a < 1 for compression, a < 0 for reflection across x-axis)
Example: Graph the quadratic function: f(x) = −(x+2)²+7
Step 1: Collect information from the function that will help us graph.
This fuction is in vertex form, so we can determine the following:
f(x) = -(x+2)² + 7
h = -2, k = 7, a < 0
axis of symmetry: x = h, x = – 2
vertex: (h,k), (-2, 7)
k: indicates vertical shift, k = 7, shift up 7 units
h: indicates horizontal shift, h = -2, shift left 2 units
a: indicates reflection , a = -1, a < 0 reflects across x-axis (opens downward)
Step 2: Starting with the parent graph, y = x², perform the transformations from Step 1. Use the axis of symmetry and vertex to check your graph.
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