Lesson 3: Graphing Parabolas

New Concept

A parabola is the U-shaped graph of a quadratic equation. We'll explore how the coefficients in $y = ax^2 + bx + c$ control the parabola's shape, direction, and position, enabling you to accurately sketch and interpret these graphs.

What’s next

This card is just the beginning. Next, you'll tackle interactive examples and practice cards to master how the coefficients $a$, $b$, and $c$ transform parabolas.

Property

The graph of the quadratic equation $y = ax^2 + bx + c$ is called a parabola. All parabolas share certain features:

• The graph has either a highest point or a lowest point, called the vertex.
• The parabola is symmetric about a vertical line, called the axis of symmetry, that runs through the vertex.
• A parabola has a $y$-intercept, and it may have zero, one, or two $x$-intercepts.
• If there are two $x$-intercepts, they are equidistant from the axis of symmetry.

Examples

• The graph of $y = x^2 - 9$ is a parabola that opens upward. Its lowest point, the vertex, is at $(0, -9)$.
• The graph of $y = -3x^2$ is a parabola that opens downward. Its highest point, the vertex, is at the origin $(0, 0)$.
• The graph of $y = x^2 - 8x + 16$ is a parabola opening upward that touches the x-axis at only one point, its vertex $(4, 0).

Property

• The parabola opens upward if $a > 0$.
• The parabola opens downward if $a < 0$.
• The magnitude of $a$ determines how wide or narrow the parabola is.
• The vertex, the $x$-intercepts, and the $y$-intercept all coincide at the origin.

Examples

• The graph of $y = 4x^2$ opens upward and is narrower than the basic parabola $y=x^2$. It passes through the points $(-1, 4)$ and $(1, 4)$.
• The graph of $y = -\frac{1}{3}x^2$ opens downward and is wider than the basic parabola. It passes through the points $(-3, -3)$ and $(3, -3)$.
• The graph of $y = 0.25x^2$ opens upward and is wider than the basic parabola. It passes through the points $(-2, 1)$ and $(2, 1).

Explanation

The coefficient 'a' acts like a stretch factor that controls the parabola's direction and width. A positive 'a' makes it open up, while a negative 'a' flips it upside down. A larger absolute value of 'a' creates a narrower parabola.

Property

Compared to the graph of $y = x^2$, the graph of $y = x^2 + c$

• is shifted upward by $c$ units if $c > 0$.
• is shifted downward by $|c|$ units if $c < 0$.

Examples

• The graph of $y = x^2 + 5$ is the basic parabola shifted 5 units up. Its vertex is located at $(0, 5)$.
• The graph of $y = x^2 - 3$ is the basic parabola shifted 3 units down. Its vertex is located at $(0, -3)$.
• The graph of $y = -x^2 + 1$ is an upside-down parabola that has been shifted 1 unit up. Its vertex is at $(0, 1).

Property

The linear term $bx$ affects the graph by shifting it horizontally, so the axis of symmetry is no longer the $y$-axis. The $x$-intercepts of the graph can be found by setting $y$ equal to zero and solving $0 = ax^2 + bx$. The $x$-coordinate of the vertex lies exactly half-way between the $x$-intercepts.

Examples

• For $y = 3x^2 - 6x$, the x-intercepts are at $x=0$ and $x=2$. The vertex's x-coordinate is $\frac{0+2}{2} = 1$. The vertex is $(1, -3)$.
• For $y = -x^2 + 5x$, the x-intercepts are at $x=0$ and $x=5$. The vertex's x-coordinate is $\frac{0+5}{2} = 2.5$. The vertex is $(2.5, 6.25)$.
• For $y = 2x^2 + 10x$, the x-intercepts are at $x=0$ and $x=-5$. The vertex's x-coordinate is $\frac{0+(-5)}{2} = -2.5$. The vertex is $(-2.5, -12.5).

Explanation

Adding a 'bx' term breaks the parabola's simple symmetry around the y-axis, pushing it left or right. To find the new center, or vertex, you can find where the graph crosses the x-axis and then locate their midpoint.

Property

For the graph of $y = ax^2 + bx$, the $x$-coordinate of the vertex is

Examples

• For the parabola $y = 2x^2 - 12x$, the vertex's x-coordinate is $x_v = \frac{-(-12)}{2(2)} = 3$. The vertex is $(3, -18)$.
• For the parabola $y = -x^2 - 6x$, the vertex's x-coordinate is $x_v = \frac{-(-6)}{2(-1)} = -3$. The vertex is $(-3, 9)$.
• For the parabola $f(x) = 0.5x^2 + 4x$, the vertex's x-coordinate is $x_v = \frac{-4}{2(0.5)} = -4$. The vertex is $(-4, -8).

Explanation

This formula is a shortcut to find the horizontal position of the vertex for any parabola. It directly calculates the x-coordinate of the axis of symmetry without needing to find the x-intercepts first. You then substitute this value back into the equation to find the y-coordinate.