Just like families share genes, families of functions share characteristics.
Recall that the function \( f(x) = x^{2} \) is the parent quadratic function. A parent function is the simplest of a family of functions. The graph of a quadratic function is a U-shaped curve called a parabola.
Review what you have learned about the parent quadratic function by completing the activity below. Answer the question on each slide, then check your answer.
This is the graph of \( f(x) = x^{2} \). The graph has several characteristics. From the graph you can see that the vertex is at (0, 0) and that the parabola opens up.
This is the graph of \( g(x) = {- x}^{2}. \)
Compare this graph to the graph of \( f(x) = x^{2} \). Explain how f(x) and g(x) are related.
Both f(x) and g(x) are parabolas. Each of them has a vertex at (0, 0). The graph of f(x) opens up, but the graph of g(x) opens down.
You can graph the parent quadratic function, or any quadratic function, by selecting values for x, substituting those values into the function, and solving for the output. Then plot the ordered pairs on the coordinate plane and connect them using a smooth curve.
Graph the parent function, \( f\left( x \right) = x^{2} \) and the function \( h\left( x \right) = 2x^{2} \) on the same coordinate plane. Then compare the characteristics of the two curves.
You can choose any values for x to substitute into each function in order to create the graph. Once you have the inputs and outputs, you can plot the ordered pairs. |
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\( f\left( x \right) = x^{2} \)
\( h\left( x \right) = 2x^{2} \)
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The graph of \( f\left( x \right) = x^{2} \) is shown in red. The graph of \( h\left( x \right) = 2x^{2} \) is shown in blue. You can see many of the features of the parabolas on the graph. Both functions open up. Both have a vertex at (0, 0). The graph of h(x) is vertically stretched compared to the graph of f(x). |
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