Introduction to Quadratic Functions | Algebra I - Grade 9

Introduction to Quadratic Functions

Description: In this lesson, you will learn about quadratic functions, their key characteristics, and how to graph them. We will go through six examples and provide practice exercises with solutions. By the end of this lesson, you'll have a strong foundation in quadratic functions.

Lesson Outline:

What is a Quadratic Function?
Standard Form of a Quadratic Function
Key Characteristics of Quadratic Functions
Graphing Quadratic Functions
Examples
Exercises
Homework
Revision

1. What is a Quadratic Function?

A quadratic function is a type of polynomial function that has the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \). The graph of a quadratic function is a parabola.

2. Standard Form of a Quadratic Function

The standard form of a quadratic function is written as \( f(x) = ax^2 + bx + c \). The value of \( a \) determines the direction of the parabola (upward if \( a > 0 \), downward if \( a < 0 \)).

3. Key Characteristics of Quadratic Functions

Quadratic functions have the following key characteristics:

Vertex: The highest or lowest point on the graph of a quadratic function.
Axis of Symmetry: A vertical line that passes through the vertex and divides the parabola into two symmetrical halves.
Y-Intercept: The point where the graph crosses the y-axis.
X-Intercepts: The points where the graph crosses the x-axis (also known as roots or zeros).

4. Graphing Quadratic Functions

To graph a quadratic function, you need to identify the vertex, axis of symmetry, and intercepts. Plot these key points and draw a smooth curve to create the parabola.

5. Examples

Example 1: Graph the quadratic function \( f(x) = x^2 - 4x + 3 \).

Solution:

The vertex is at \( (2, -1) \), and the x-intercepts are at \( x = 1 \) and \( x = 3 \). The graph is a parabola opening upward. Graph of quadratic function f(x) = x^2 - 4x + 3

Example 2: Find the vertex of the quadratic function \( f(x) = -2x^2 + 4x + 1 \).

Solution:

The vertex can be found using the formula \( x = -\frac{b}{2a} \). For this function, the vertex is at \( (1, 3) \). Graph showing the vertex of f(x) = -2x^2 + 4x + 1

Example 3: Find the x-intercepts of the quadratic function \( f(x) = x^2 - 5x + 6 \).

Solution:

Solving \( x^2 - 5x + 6 = 0 \), we find the x-intercepts at \( x = 2 \) and \( x = 3 \).

Example 4: Identify the axis of symmetry for \( f(x) = 3x^2 + 6x - 9 \).

Solution:

The axis of symmetry is given by the equation \( x = -\frac{b}{2a} \). For this function, the axis of symmetry is \( x = -1 \).

Example 5: Graph the quadratic function \( f(x) = -x^2 + 4x - 3 \).

Solution:

The vertex is at \( (2, 1) \), and the graph is a parabola opening downward. Graph of quadratic function f(x) = -x^2 + 4x - 3

Example 6: Find the y-intercept of the quadratic function \( f(x) = 2x^2 - 4x + 1 \).

Solution:

The y-intercept occurs when \( x = 0 \), so the y-intercept is \( f(0) = 1 \).

6. Exercises

Exercise 1: Find the vertex of the quadratic function \( f(x) = x^2 - 6x + 5 \).

Solution: The vertex is at \( (3, -4) \).

Exercise 2: Find the x-intercepts of the quadratic function \( f(x) = 2x^2 - 8x + 6 \).

Solution: The x-intercepts are at \( x = 1 \) and \( x = 3 \).

Exercise 3: Identify the axis of symmetry for the quadratic function \( f(x) = -x^2 + 2x + 4 \).

Solution: The axis of symmetry is at \( x = 1 \).

7. Homework

Solve the following problems and submit your answers in the comments section below:

Find the vertex of the function \( f(x) = 2x^2 - 4x + 1 \).
Graph the quadratic function \( f(x) = x^2 + 2x - 3 \).
Find the x-intercepts of the quadratic function \( f(x) = -x^2 + 4x - 4 \).

8. Revision

Review the key concepts of quadratic functions, including the standard form, vertex, axis of symmetry, and intercepts. Practice graphing different quadratic functions and identifying their key characteristics.