MTH 065 Elementary Algebra II Chapter 11 Quadratic Functions and Equations Section 11.6 Quadratic Functions and Their Graphs

Quadratic Functions Graphs … U-Shaped curves called Parabolas Vertex: Point at the bottom/top of the U Axis of Symmetry: Vertical line through the vertex. y-intercept: f(0) x-intercepts: f(x) = 0 (“zeros” of the function) or a, b, & c are real constants with a ≠ 0.

f(x) = ax 2 Case 1: a = 1 f(x) = x 2 xf(x) –1–11 24 –2–24

f(x) = ax 2 Case 2: a = –1 f(x) = –x 2 xf(x) 00 1–1–1 –1–1–1–1 2–4–4 –2–2–4–4 If a > 0, the parabola opens upward. If a < 0, the parabola opens downward.

f(x) = ax 2 Case 3: a > 1 f(x) = 3x 2 xf(x) –1– –2–2

f(x) = ax 2 Case 4: 0 < a < 1 f(x) = ⅛x 2 xf(x) 00 1⅛ –1–1⅛ 2½ –2–2½ 2 points determine a unique line. 3 points determine a unique parabola.

f(x) = ax 2 Summary … Vertex: (0, 0) Axis of Symmetry: x = 0 Opens up: a > 0 Opens down: a < 0 Narrow: |a| > 1 Wide: |a| < 1 Quick graph … Plot the vertex (i.e. the origin). Find/plot a second point. Use symmetry to draw the graph.

f(x) = ax 2 + k What happens if a constant is added? f(x) = x is the same as f(x) = x 2 moved 2 units up

f(x) = ax 2 – k What happens if a constant is subtracted? f(x) = x 2 – 2 is the same as f(x) = x 2 moved 2 units down

f(x) = ax 2 ± k Summary … Vertex: (0, k) w/+; (0, –k) w/– Axis of Symmetry: x = 0 a … same as before Quick graph … Plot the vertex. Find/plot a second point. Use symmetry to draw the graph.

f(x) = a(x – h) 2 What happens if a constant is subtracted from x? f(x) = (x – 3) 2 is the same as f(x) = x 2 moved 3 units to the right

f(x) = a(x + h) 2 What happens if a constant is added to x? f(x) = (x + 3) 2 is the same as f(x) = x 2 moved 3 units to the left

Summary … Vertex: (h, 0) w/–; (–h, 0) w/+ Axis of Symmetry: x = h w/–; x = –h w/+; a … same as before Quick graph … Plot the vertex. Find/plot the y-intercept. Use symmetry to draw the graph.

f(x) = a(x – h) 2 + k What happens with both parameters? f(x) = (x – 3) is the same as f(x) = x 2 moved 3 units to the right and 2 units up

Summary … Vertex: (h, k) w/–+; (–h, k) w/++ (h, –k) w/––; (–h, –k) w/+– Axis of Symmetry: x = h w/–+ or –– x = –h w/++ or +– a … same as before Quick graph … Plot the vertex. Find/plot the y-intercept. Use symmetry to draw the graph.

What is the equation of the graph? Vertex: (–2, –1) f(x) = a(x + 2) 2 – 1 y-intercept: (0, 1) 1 = a(0 + 2) 2 – 1 1 = 4a – 1 a = ½ Equation is …