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.
Terms introduced in this lesson:
general form, standard form of a quadratic function
vertical, horizontal shift
x−intercepts of a parabola
vertex of a parabola
symmetric, line of symmetry
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:
- A maximum of two x−intercepts
- 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 y−values which can make a y−axis too big. See Examples 1a and 1c.
Use Example 2 to introduce intercept form.
Solution: Write the quadratic function in intercept form by factoring the right side.
- 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.