Lesson 4: Graphing Quadratic Equations

New Concept

This lesson introduces a powerful method for graphing parabolas. Instead of just plotting points, you'll learn to find key features—the $x$-intercepts, the $y$-intercept, and the vertex—to sketch any quadratic equation quickly and accurately.

What’s next

Next, you'll use interactive examples to master finding a parabola's intercepts and vertex. Then, you will apply these new skills in a series of practice cards.

Property

To find the $x$-intercepts of the graph of

we set $y = 0$ and solve the equation

Property

The high or low point of a parabola is called its vertex. Because of the symmetry of a parabola, the $x$-intercepts are located at equal distances on either side of the axis of symmetry, which passes through the vertex.

Examples

• If a parabola has $x$-intercepts at $(-1, 0)$ and $(5, 0)$, its axis of symmetry must be at $x=2$, exactly halfway between them. The vertex lies on this line.

• For a parabola that opens upward, like $y=x^2$, the vertex at $(0,0)$ is the minimum point on the entire graph.

Property

1. The $x$-coordinate of the vertex is the average of the $x$-intercepts.

2. To find the $y$-coordinate of the vertex, substitute its $x$-coordinate into the equation of the parabola.

Examples

• For $y = x^2 - 6x$, the intercepts are at $x=0$ and $x=6$. The vertex's $x$-coordinate is $\frac{0+6}{2} = 3$. The $y$-coordinate is $3^2 - 6(3) = 9 - 18 = -9$. The vertex is $(3, -9)$.

Property

1. Find the $x$-intercepts: set $y = 0$ and solve for $x$.

2. Find the vertex: the $x$-coordinate is the average of the $x$-intercepts. Find the $y$-coordinate by substituting the $x$-coordinate into the equation of the parabola.

3. Find the $y$-intercept: set $x = 0$ and solve for $y$.