Learn on PengiCalifornia Reveal Math, Algebra 1Unit 10: Quadratic Functions

10-1 Graphing Quadratic Functions

In this Grade 9 lesson from California Reveal Math Algebra 1, students learn to analyze and graph quadratic functions written in standard form f(x) = ax² + bx + c by identifying the axis of symmetry, vertex, and y-intercept of a parabola. Students practice using the formula x = -b/2a to find the axis of symmetry, determine whether the vertex is a minimum or maximum based on the value of a, and describe end behavior. Real-world applications, including suspension bridge design and a horse's jump trajectory, illustrate how quadratic functions model maximum and minimum points.

Section 1

Quadratic Function

Property

The standard form of a quadratic function is given by the equation $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are real numbers and $a \neq 0$ .

Examples

The function $f(x) = 3x^2 + 6x - 2$ is a quadratic function where $a=3$ , $b=6$ , and $c=-2$ .
The function $g(x) = -x^2 + 5$ is a quadratic function where $a=-1$ , $b=0$ , and $c=5$ .
The function $h(x) = x^2 + 4x$ is a quadratic function where $a=1$ , $b=4$ , and $c=0$ .

Explanation

A quadratic function creates a U-shaped graph called a parabola. Unlike linear functions that form straight lines, these functions include a squared term ( $x^2$ ), which creates the distinctive curve. The values of $a$ , $b$ , and $c$ determine the parabola's shape and position.

Section 2

Applying the Negative Sign in the Axis of Symmetry Formula

Property

For a quadratic function in standard form $f(x) = ax^2 + bx + c$ , the axis of symmetry is:

x = -\frac{b}{2a}

Section 3

Intercepts and Symmetry Points of a Quadratic Function

Property

For a quadratic function in standard form $f(x) = ax^2 + bx + c$ :

Y-intercept: Set $x = 0$ to get the point $(0,\, c)$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Quadratic Function

Property

The standard form of a quadratic function is given by the equation $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are real numbers and $a \neq 0$ .

Examples

The function $f(x) = 3x^2 + 6x - 2$ is a quadratic function where $a=3$ , $b=6$ , and $c=-2$ .
The function $g(x) = -x^2 + 5$ is a quadratic function where $a=-1$ , $b=0$ , and $c=5$ .
The function $h(x) = x^2 + 4x$ is a quadratic function where $a=1$ , $b=4$ , and $c=0$ .

Explanation

Section 2

Applying the Negative Sign in the Axis of Symmetry Formula

Property

For a quadratic function in standard form $f(x) = ax^2 + bx + c$ , the axis of symmetry is:

x = -\frac{b}{2a}

Section 3

Intercepts and Symmetry Points of a Quadratic Function

Property

For a quadratic function in standard form $f(x) = ax^2 + bx + c$ :

Y-intercept: Set $x = 0$ to get the point $(0,\, c)$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Quadratic Function

Property

The standard form of a quadratic function is given by the equation $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are real numbers and $a \neq 0$ .

Examples

The function $f(x) = 3x^2 + 6x - 2$ is a quadratic function where $a=3$ , $b=6$ , and $c=-2$ .
The function $g(x) = -x^2 + 5$ is a quadratic function where $a=-1$ , $b=0$ , and $c=5$ .
The function $h(x) = x^2 + 4x$ is a quadratic function where $a=1$ , $b=4$ , and $c=0$ .

Explanation

Section 2

Applying the Negative Sign in the Axis of Symmetry Formula

Property

For a quadratic function in standard form $f(x) = ax^2 + bx + c$ , the axis of symmetry is:

x = -\frac{b}{2a}

Section 3

Intercepts and Symmetry Points of a Quadratic Function

Property

For a quadratic function in standard form $f(x) = ax^2 + bx + c$ :

Y-intercept: Set $x = 0$ to get the point $(0,\, c)$ .