Chapter 2: Quadratic Functions

*Lesson 1: Transformations of Quadratic Functions*Quadratic Functions

A quadratic function is a function that can be written in the form f(x) = a(x - h)2 + k, where a ≠ 0

The U-shaped graph of a quadratic function is called a parabola

Translations

Horizontal Translation

f(x) = x2

f(x - h) = (x - h)2

Shifts left when h < 0

Shifts right when h > 0

Vertical Translation

f(x) = x2

f(x) + k = x2 + k

Shifts down when k < 0

Shifts up when k > 0

Reflections

Reflection in the x-axis

f(x) = x2

-f(x) = -(x2) = -x2

Reflection in the y-axis

f(x) = x2

f(-x) = (-x)2 = x2

Stretches and Shrinks

Horizontal Stretches and Shrinks

f(x) = x2

f(ax) = (ax)2

Horizontal stretch (away from y-axis) when 0 < a < 1

Horizontal shrink (towards y-axis) when a > 1

Vertical Stretches and Shrinks

f(x) = x2

a * f(x) = ax2

Vertical stretch (away from x-axis) when a > 1

Vertical shrink (towards x-axis) when 0 < a < 1

Vertex Form

The lowest part on a parabola that opens up or the highest point on a parabola that opens down is called the vertex

The vertex form of a quadratic function is

__f(x) = a(x - h)2 + k__, where a ≠ 0 and the__vertex is (h, k)__

*Lesson 2: Characteristics of Quadratic Functions*Quadratic Function Forms

Axis of Symmetry - a line that divides a parabola into mirror images and passes through the vertex; the axis of symmetry is the vertical line x = h or x = -b/2a

Standard Form - a quadratic function written in the form f(x) = ax2 + bx + c

Intercept Form - a quadratic function in the form f(x) = a(x - p)(x - q)

Properties of the Standard Form

The parabola opens up when a > 0 and opens down when a < 0

The graph is narrower than the graph of f(x) = x2 when |a| > 1 and wider when |a| < 1

The axis of symmetry is x = -b/2a and the vertex is (-b/2a, f(-b/2a))

The y-intercept is c; the point (0, c) is on the parabola

Minimum and Maximum Values

For the quadratic function f(x) = ax2 + bx + c, the y-coordinate of the vertex is the minimum value of the function when a > 0 and the maximum value when a < 0

a > 0

Minimum value: f(-b/2a)

Domain: all real numbers

Range: y ≥ f(-b/2a)

Decreasing to the left of x = -b/2a

Increasing to the right of x = -b/2a

a < 0

Maximum value: f(-b/2a)

Domain: all real numbers

Range: y ≤ f(-b/2a)

Increasing to the left of x = -b/2a

Decreasing to the right of x = -b/2a

Properties of the Intercept Form

Because f(p) = 0 and f(q) = 0, p and q are the x-intercepts of the graph of the function

The axis of symmetry is halfway between (p, 0) and (q, 0); x = (p + q)/2

The parabola opens up when a > 0 and opens down when a < 0

*Lesson 3: Modeling with Quadratic Equations*Writing Quadratic Equations

Given a point and the vertex (h, k), use vertex form: y = a(x - h)2 + k

Given a point and x-intercepts p and q, use intercept form y = a(x - p)(x - q)

Given three points, write and solve a system of three equations in three variables

*Lesson 1: Transformations of Quadratic Functions*Quadratic Functions

A quadratic function is a function that can be written in the form f(x) = a(x - h)2 + k, where a ≠ 0

The U-shaped graph of a quadratic function is called a parabola

Translations

Horizontal Translation

f(x) = x2

f(x - h) = (x - h)2

Shifts left when h < 0

Shifts right when h > 0

Vertical Translation

f(x) = x2

f(x) + k = x2 + k

Shifts down when k < 0

Shifts up when k > 0

Reflections

Reflection in the x-axis

f(x) = x2

-f(x) = -(x2) = -x2

Reflection in the y-axis

f(x) = x2

f(-x) = (-x)2 = x2

Stretches and Shrinks

Horizontal Stretches and Shrinks

f(x) = x2

f(ax) = (ax)2

Horizontal stretch (away from y-axis) when 0 < a < 1

Horizontal shrink (towards y-axis) when a > 1

Vertical Stretches and Shrinks

f(x) = x2

a * f(x) = ax2

Vertical stretch (away from x-axis) when a > 1

Vertical shrink (towards x-axis) when 0 < a < 1

Vertex Form

The lowest part on a parabola that opens up or the highest point on a parabola that opens down is called the vertex

The vertex form of a quadratic function is

__f(x) = a(x - h)2 + k__, where a ≠ 0 and the__vertex is (h, k)__

*Lesson 2: Characteristics of Quadratic Functions*Quadratic Function Forms

Axis of Symmetry - a line that divides a parabola into mirror images and passes through the vertex; the axis of symmetry is the vertical line x = h or x = -b/2a

Standard Form - a quadratic function written in the form f(x) = ax2 + bx + c

Intercept Form - a quadratic function in the form f(x) = a(x - p)(x - q)

Properties of the Standard Form

The parabola opens up when a > 0 and opens down when a < 0

The graph is narrower than the graph of f(x) = x2 when |a| > 1 and wider when |a| < 1

The axis of symmetry is x = -b/2a and the vertex is (-b/2a, f(-b/2a))

The y-intercept is c; the point (0, c) is on the parabola

Minimum and Maximum Values

For the quadratic function f(x) = ax2 + bx + c, the y-coordinate of the vertex is the minimum value of the function when a > 0 and the maximum value when a < 0

a > 0

Minimum value: f(-b/2a)

Domain: all real numbers

Range: y ≥ f(-b/2a)

Decreasing to the left of x = -b/2a

Increasing to the right of x = -b/2a

a < 0

Maximum value: f(-b/2a)

Domain: all real numbers

Range: y ≤ f(-b/2a)

Increasing to the left of x = -b/2a

Decreasing to the right of x = -b/2a

Properties of the Intercept Form

Because f(p) = 0 and f(q) = 0, p and q are the x-intercepts of the graph of the function

The axis of symmetry is halfway between (p, 0) and (q, 0); x = (p + q)/2

The parabola opens up when a > 0 and opens down when a < 0

*Lesson 3: Modeling with Quadratic Equations*Writing Quadratic Equations

Given a point and the vertex (h, k), use vertex form: y = a(x - h)2 + k

Given a point and x-intercepts p and q, use intercept form y = a(x - p)(x - q)

Given three points, write and solve a system of three equations in three variables