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10.1: Graphs of Quadratic Functions

Difficulty Level: 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
xintercepts 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 xintercepts
  • 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 y=x2.

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 yvalues which can make a yaxis too big. See Examples 1a and 1c.

Use Example 2 to introduce intercept form.

Additional Examples:

Find the xintercepts and vertices of the following quadratic functions.

a. y=x2+3x10

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

y=x2+3x10=(x+5)(x2)

Set y=0. So, the xintercepts are (5,0) and (2,0). The xvalue of the vertex is halfway between the two xintercepts:

5+22=15

To the yvalue, plug in the xvalue into the original equation:

y=(1.5)2+3(1.5)10=12.25

The vertex is (1.5,12.25).

b.y=2x2+56x+120

Answer: The xintercepts are (2,0) and (30,0) and vertex is (14,512).

c. y=x2+15x36

Answer: The xintercepts are (12,0) and (30,0) and vertex is (7.5,20.25).

Point out that the vertex lies on the line of symmetry; therefore, xintercepts are reflections of each other and the vertex is halfway between them.

By analyzing the effects of the coefficients a and c in y=ax2+c, 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.

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