Lesson 9.6: Graph Quadratic Functions Using Properties

New Concept

Explore quadratic functions of the form $f(x) = ax^2 + bx + c$. You'll learn to identify key properties like the vertex and intercepts to precisely graph parabolas and find their maximum or minimum values.

What’s next

Up next, you'll master finding a parabola's vertex and intercepts through guided examples and targeted practice cards, preparing you to graph any quadratic function.

Property

A quadratic function, where $a$, $b$, and $c$ are real numbers and $a \neq 0$, is a function of the form

We call the graph of a quadratic function a parabola.

Property

For the graph of the quadratic function $f(x) = ax^2 + bx + c$, if

• $a > 0$, the parabola opens upward $\uparrow$
• $a < 0$, the parabola opens downward $\downarrow$

Examples

• For $f(x) = 5x^2 - 2x + 10$, the parabola opens upward because $a=5$, which is positive.
• For $f(x) = -2x^2 + 8x$, the parabola opens downward because $a=-2$, which is negative.
• For $f(x) = \frac{1}{3}x^2 - 1$, the parabola opens upward because $a=\frac{1}{3}$, which is positive.

Explanation

The sign of the leading coefficient, $a$, controls which way the parabola opens. Think of it this way: a positive 'a' is like a smile (opens up), while a negative 'a' is like a frown (opens down).

Property

The graph of the function $f(x) = ax^2 + bx + c$ is a parabola where:

• the axis of symmetry is the vertical line $x = -\frac{b}{2a}$.
• the vertex is a point on the axis of symmetry, so its $x$-coordinate is $-\frac{b}{2a}$.
• the $y$-coordinate of the vertex is found by substituting $x = -\frac{b}{2a}$ into the quadratic equation.

Examples

• For $f(x) = x^2 - 6x + 11$, the axis of symmetry is $x = -\frac{-6}{2(1)} = 3$. The vertex is $(3, f(3))$, which is $(3, 2)$.
• For $f(x) = -2x^2 - 8x - 5$, the axis of symmetry is $x = -\frac{-8}{2(-2)} = -2$. The vertex is $(-2, f(-2))$, which is $(-2, 3)$.
• For $f(x) = 4x^2 - 8$, the axis of symmetry is $x = -\frac{0}{2(4)} = 0$. The vertex is $(0, f(0))$, which is $(0, -8)$.

Explanation

The axis of symmetry is an invisible vertical line that splits the parabola into two perfect mirror images. The vertex is the parabola's turning point (either the very bottom or very top), and it always sits right on this line.

Property

To find the intercepts of a parabola whose function is $f(x) = ax^2 + bx + c$:

Examples

• For $f(x) = x^2 - 3x - 10$, the $y$-intercept is $(0, -10)$. To find the $x$-intercepts, we solve $x^2 - 3x - 10 = 0$, which factors to $(x-5)(x+2)=0$. The $x$-intercepts are $(5, 0)$ and $(-2, 0)$.
• For $f(x) = x^2 + 4x + 4$, the $y$-intercept is $(0, 4)$. The equation $x^2 + 4x + 4 = 0$ factors to $(x+2)^2=0$, so the only $x$-intercept is $(-2, 0)$.
• For $f(x) = 2x^2 + x + 3$, the $y$-intercept is $(0, 3)$. The discriminant $b^2 - 4ac = 1^2 - 4(2)(3) = -23$ is negative, so there are no $x$-intercepts.

Property

To graph a quadratic function using properties:
Step 1. Determine whether the parabola opens upward or downward.
Step 2. Find the equation of the axis of symmetry.
Step 3. Find the vertex.
Step 4. Find the $y$-intercept. Find the point symmetric to the $y$-intercept across the axis of symmetry.
Step 5. Find the $x$-intercepts. Find additional points if needed.
Step 6. Graph the parabola.

Examples

• To sketch $f(x) = x^2 + 2x - 8$: it opens up ($a=1$), axis is $x=-1$, vertex is $(-1, -9)$, $y$-intercept is $(0, -8)$, and $x$-intercepts are $(2, 0)$ and $(-4, 0)$.
• To sketch $f(x) = -x^2 + 6x - 9$: it opens down ($a=-1$), axis is $x=3$, vertex is $(3, 0)$, $y$-intercept is $(0, -9)$, and the only $x$-intercept is $(3, 0)$.
• To sketch $f(x) = x^2+2x+3$: it opens up ($a=1$), axis is $x=-1$, vertex is $(-1, 2)$, $y$-intercept is $(0, 3)$, and there are no $x$-intercepts.

Explanation

Instead of plotting tons of points, you can sketch an accurate parabola by following this step-by-step recipe. Finding these key features gives you a complete blueprint of the graph, showing its direction, turning point, and where it crosses the axes.

Property

The $y$-coordinate of the vertex of the graph of a quadratic function is the

• minimum value of the quadratic equation if the parabola opens upward.
• maximum value of the quadratic equation if the parabola opens downward.

Examples

• The function $f(x) = 2x^2 + 8x + 1$ opens upward and has a minimum value. The vertex is at $x=-2$, so the minimum value is $f(-2) = -7$.
• The function $f(x) = -x^2 - 4x + 6$ opens downward and has a maximum value. The vertex is at $x=-2$, so the maximum value is $f(-2) = 10$.
• A rocket's height is modeled by $h(t) = -16t^2 + 96t$. It opens downward, so it has a maximum height. The vertex occurs at $t=3$ seconds, and the maximum height is $h(3) = 144$ feet.

Explanation

The vertex isn't just a point; its $y$-value tells you the absolute lowest or highest value the function can ever reach. This is extremely useful for solving real-world problems where you need to find the best possible outcome.