Graphing Parabolas

Section 3.3 Graphing Parabolas

In this Lesson we consider the effects of the parameters \(a\text{,}\) \(b\text{,}\) and \(c\) on the graph of the basic parabola. You could have students work on the Activities in groups, stopping to summarize after each part.

Activity 3.3.1. Graphing Practice I.

Complete the table of values, then sketch the graph on the grid with the basic parabola and label the coordinates of three points on the graph.

Students have trouble evaluating even these simple quadratic expressions. A common error, for example, is to enter \(-4^2\) into the calculator when evaluating \(x^2\) at \(-4\text{.}\)

i. \(~y=\frac{1}{2}x^2\)

\(x\) \(-2\) \(-1\) \(0\) \(1\) \(2\)

\(y\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)

ii. \(~y=3x^2\)

\(x\) \(-2\) \(-1\) \(0\) \(1\) \(2\)

\(y\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)
Compare each graph to the basic parabola, \(y=x^2\text{.}\) How is it different?

Summarize the results and direct students’ attention to the appropriate page in the textbook.

Activity 3.3.2. Graphing Practice II.

Complete the table of values, then sketch the graph on the grid with the basic parabola and label the coordinates of three points on the graph.

i. \(~y=x^2 - 9\)

\(x\) \(-4\) \(-3\) \(-1\) \(0\) \(1\) \(0\) \(1\)

\(y\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\)

ii. \(~y=9 - x^2\)

\(x\) \(-4\) \(-3\) \(-1\) \(0\) \(1\) \(0\) \(1\)

\(y\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\)
Find the \(x\)-intercepts of each graph.

Find the vertex of each graph.
Compare each graph to the basic parabola, \(y=x^2\text{.}\) How is it different?

Note the symmetry of each graph and the location of the vertex. Again direct students’ attention to the appropriate page in the textbook.

Activity 3.3.3. Graphing Practice III.

Complete the table of values, then sketch the graph on the grid with the basic parabola and label the coordinates of three points on the graph.

i. \(~y=x^2 + 6x\)

\(x\)	\(-6\)	\(-4\)	\(-3\)	\(-2\)	\(0\)	\(1\)
\(y\)	\(\hphantom{000}\)	\(\hphantom{000}\)	\(\hphantom{000}\)	\(\hphantom{000}\)	\(\hphantom{000}\)	\(\hphantom{000}\)

ii. \(~y=2x^2 - 6x\)

\(x\)	\(-1\)	\(0\)	\(1\)	\(\frac{3}{2}\)	\(2\)	\(3\)	\(4\)
\(y\)	\(\hphantom{000}\)	\(\hphantom{000}\)	\(\hphantom{000}\)	\(\hphantom{000}\)	\(\hphantom{000}\)	\(\hphantom{000}\)	\(\hphantom{000}\)

Find the \(x\)-intercepts of each graph.

Help students find the \(x\)-intercepts, pointing out the use of factoring instead of extracting roots to solve the equations.

Show the class how to find the \(x\)-coordinate of the vertex as the average of the \(x\)-intercepts. They will also need to be prompted on how to find the \(y\)-coordinate of the vertex.
Find the vertex of each graph.
Compare each graph to the basic parabola, \(y=x^2\text{.}\) How is it different?

Activity 3.3.4. Intercepts and Vertex.

1. How do we find the \(y\)-intercept of a parabola from its equation?
2. How do we find the \(x\)-intercepts of a parabola from its equation?
3. Recall that to find the average of two numbers we add them up, and divide the result by 2.
1. If the \(x\)-intercepts of a parabola are \(x=p\) and \(x=q\text{,}\) what is the \(x\)-coordinate of its vertex?
2. If the \(x\)-intercepts of a parabola are \(x=0\) and \(x=\dfrac{-b}{a}\text{,}\) what is the \(x\)-coordinate of its vertex?
3. How do we find the \(y\)-coordinate of the vertex from the equation?
  
  This Activity leads students to the formula for the \(x\)-coordinate of the vertex. After deriving the formula, demonstrate that it works on the graphs in the prevous Activity.
Describe how each of the three parameters \(a\text{,}\) \(b\text{,}\) and \(c\) affect the graph of the basic parabola.
1. \(\displaystyle y=ax^2\)
2. \(\displaystyle y=x^2 + c\)
3. \(\displaystyle y=x^2 + bx\)

Subsection 3.3.1 Check Your Understanding

What are the \(x\)-intercepts of the graph of \(~y=x^2-16\text{?}\) What are the \(x\)-intercepts of the graph of \(y=x^2-16x\text{?}\)
What is the axis of symmetry of the graph of \(y=x^2-16\text{?}\) What is the axis of symmetry of the graph of \(~y=x^2-16\text{?}\)
How is the graph of \(~y=2x^2+5~\) different from the basic parabola? What are the coordinates of its vertex?
Compare the graphs of \(y=x^2-25\) and \(y=25-x^2\text{.}\)

Subsection 3.3.2 Wrap Up

In this Lesson, we worked on the following skills and goals related to quadratic models:

Describe the effect of each of the parameters \(a\text{,}\) \(b\text{,}\) and \(c\) on the graph of a parabola
Find the \(x\)- and \(y\)-intercepts of a parabola
Find the vertex of a parabola
Sketch the graph of a parabola

Subsection 3.3.3 Questions for Writing or Discussion

Describe what the parameter \(a\) tells you about the graph of \(y=ax^2\text{.}\)
Explain why the \(x\)-coordinate of the vertex is the average of the \(x\)-intercepts of the graph.
Describe the differences between the graphs of \(y=x^2-4\) and \(y=x^2-4x\text{.}\)
Explain why the graph of \(y=x^2-2\) is shifted down 2 units compared to the basic parabola.

Concept Questions.

Which point on a parabola always lies on the axis of symmetry?
1. The \(x\)-intercept
2. The \(y\)-intercept
3. The vertex
4. The origin
The \(x\)-intercept of the graph of \(~y=ax^2+bx+c~\) is the same as the value of which parameter?
1. \(\displaystyle a\)
2. \(\displaystyle b\)
3. \(\displaystyle c\)
4. None of these
What does the value of \(a\) tell us about the graph of \(~y=ax^2+bx+c~\text{?}\)
1. The \(y\)-intercept
2. The number of \(x\)-intercepts
3. The \(x\)-coordinate of the vertex
4. The width of the parabola
Give the equation of a parabola that has no \(x\)-intercepts.
1. \(\displaystyle y=x^2\)
2. \(\displaystyle y=x^2+4\)
3. \(\displaystyle y=x^2-4\)
4. \(\displaystyle y=x^2+x\)

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