D. The Effect of a, p, and q
Completion requirements
D. The Effect of a, p, and q
| Investigation |
Effect of \(a\)
Using your graphing calculator, or another graphing program, such as Desmos, graph the following functions. Compare the other functions to the basic quadratic function, \(f(x) = x^2 \). Use the tables below to investigate the effect of \(a\) on the graph of the function.
Looking over the table, think about any conclusions you could make regarding the values of \(a\) and \(q\) with respect to the direction of opening, stretching, domain/range, and the number of \(x\)-intercepts.
| Function |
|
\(f(x) = x^2 \) | \(g(x) = -x^2 \) |
|---|---|---|---|
\(f(x) = x^2 \) and \(g(x) = -x^2 \)
|
Value of \(a\) |
|
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| Direction of Opening |
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Narrower or wider compared to
\(f(x) = x^2 \) |
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| Value of \(q\) |
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| Domain |
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| Range |
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| Number of \(x\)-intercepts |
|
|
| Function |
|
\(f(x) = x^2 + 2\) | \(g(x) = -x^2 + 2\) |
|---|---|---|---|
|
\(f(x) = x^2 + 2\) and \(g(x) = -x^2 + 2 \)
|
Value of \(a\) |
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|
| Direction of Opening |
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Narrower or wider compared to
\(f(x) = x^2 \) |
|
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| Value of \(q\) |
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| Domain |
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| Range |
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| Number of \(x\)-intercepts |
|
|
| Function |
|
\(f(x) = x^2 - 2\) | \(g(x) = -x^2 - 2\) |
|---|---|---|---|
|
\(f(x) = x^2 - 2\) and \(g(x) = -x^2 - 2\)
|
Value of \(a\) |
|
|
| Direction of Opening |
|
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|
|
Narrower or wider compared to
\(f(x) = x^2 \) |
|
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| Value of \(q\) |
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| Domain |
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| Range |
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|
| Number of \(x\)-intercepts |
|
|
| Function |
|
\(f(x) = 2x^2 \) | \(g(x) = -2x^2 \) |
|---|---|---|---|
|
\(f(x) = 2x^2 \) and \(g(x) = -2x^2 \)
|
Value of \(a\) |
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|
| Direction of Opening |
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|
|
Narrower or wider compared to
\(f(x) = x^2 \) |
|
|
|
| Value of \(q\) |
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|
| Domain |
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| Range |
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| Number of \(x\)-intercepts |
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| Function | \(f(x) = 0.5x^2 \) | \(g(x) = -0.5x^2 \) | |
|---|---|---|---|
|
\(f(x) = 0.5x^2 \) and \(g(x) = -0.5x^2 \)
|
Value of \(a\) | ||
| Direction of Opening | |||
|
Narrower or wider compared to
\(f(x) = x^2 \) |
|||
| Value of \(q\) | |||
| Domain | |||
| Range | |||
| Number of \(x\)-intercepts |
Looking over the table, think about any conclusions you could make regarding the values of \(a\) and \(q\) with respect to the direction of opening, stretching, domain/range, and the number of \(x\)-intercepts.