# 10.1: Graphs of Quadratic Functions

**At Grade**Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

- Graph quadratic functions.
- Compare graphs of quadratic functions.
- Graph quadratic functions in intercept form.
- Analyze graphs of real-world quadratic functions.

## Vocabulary

Terms introduced in this lesson:

- quadratic equations
- quadratic functions
- parabola
- smooth curve
- general form, standard form of a quadratic function
- coefficients
- orientation
- dilation
- vertical, horizontal shift
- \begin{align*}x-\end{align*}intercepts of a parabola
- vertex of a parabola
- symmetric, line of symmetry
- quadratic expression
- intercept form

## Teaching Strategies and Tips

Point out that students frequently come into contact with parabolas.

- The path of any projectile is part of a parabola.
- Cross-sections of satellite dishes and flashlight mirrors are parabolas.
- See introduction for more examples.

Draw several parabolas and lead students in discovering the essential features:

- Symmetry
- A maximum of two \begin{align*}x-\end{align*}intercepts
- Vertices
- None of these features are shared by the other graphs students have been studying.

Use a table of values to show *why* the parabola has a U-shape.

- Allow students to observe the effect of squaring on negative numbers.
- Use a basic quadratic function such as \begin{align*}y=x^2\end{align*}.

When graphing quadratic functions, suggest that:

- Sketches do not need to be perfect, but curves should be U-shaped rather than V-shaped. In general, join points with a smooth curve rather than with segments.
- Plot more points until the familiar curve is in view. More points are necessary to draw a parabola accurately than to graph lines, since a parabola is curved. See Example 1c.
- Knowing the vertex cuts down on the number of points to be plotted, because that is where the parabola opens. Advise students to choose at least one point on either side of the vertex.
- Not all points in a table must be plotted. This is especially true for points with large \begin{align*}y-\end{align*}values which can make a \begin{align*}y-\end{align*}axis too big. See Examples 1a and 1c.

Use Example 2 to introduce *intercept form*.

Additional Examples:

*Find the* \begin{align*}x-\end{align*}*intercepts and vertices of the following quadratic functions.*

a. \begin{align*}y=x^2+3x-10\end{align*}

Solution: Write the quadratic function in intercept form by factoring the right side.

\begin{align*}y=x^2+3x-10=(x+5)(x-2)\end{align*}

Set \begin{align*}y = 0\end{align*}. So, the \begin{align*}x-\end{align*}intercepts are \begin{align*}(-5,0)\end{align*} and \begin{align*}(2,0)\end{align*}. The \begin{align*}x-\end{align*}value of the vertex is halfway between the two \begin{align*}x-\end{align*}intercepts:

\begin{align*}\frac{-5+2}{2} =-15\end{align*}

To the \begin{align*}y-\end{align*}value, plug in the \begin{align*}x-\end{align*}value into the original equation:

\begin{align*}y=(-1.5)^2 +3(-1.5)-10=-12.25\end{align*}

The vertex is \begin{align*}(-1.5,-12.25)\end{align*}.

b.\begin{align*}y=-2x^2+56x+120\end{align*}

Answer: The \begin{align*}x-\end{align*}intercepts are \begin{align*}(-2, 0)\end{align*} and \begin{align*}(30, 0)\end{align*} and vertex is \begin{align*}(14, 512)\end{align*}.

c. \begin{align*}y=-x^2+15x-36\end{align*}

Answer: The \begin{align*}x-\end{align*}intercepts are \begin{align*}(12,0)\end{align*} and \begin{align*}(30,0)\end{align*} and vertex is \begin{align*}(7.5, 20.25)\end{align*}.

Point out that the vertex lies on the line of symmetry; therefore, \begin{align*}x-\end{align*}intercepts are reflections of each other and the vertex is halfway between them.

By analyzing the effects of the coefficients \begin{align*}a\end{align*} and \begin{align*}c\end{align*} in \begin{align*}y=ax^2+c\end{align*}, teachers can introduce the basic transformations: shift, stretch, and flip. Ask:

- What makes the graph of the parabola open up? Down?
- What makes a parabola wider? Narrower?
- How can the vertex of one parabola be placed higher than another?

Use graphing calculators to:

- Find intercepts, vertices, and points of intersection to any degree of precision.
- Compare several quadratic functions simultaneously.

## Error Troubleshooting

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