Question Statement

Draw the graph of the parent function $y=x^2$ and then sketch the graphs of the following functions using transformations. Verify the results using a graphical calculator:

(a) $y=(3x{)}^2$ (b) $y=(-3x{)}^2$ (c) $y={\left(\frac{x}{3}\right)}^2$ (d) $y={\left(-\frac{x}{3}\right)}^2$

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Background and Explanation

To sketch these graphs, we use the concept of horizontal transformations. For a function $y=f(kx)$, if $|k|>1$, the graph is horizontally compressed toward the y-axis by a factor of $\frac{1}{k}$, and if $0<|k|<1$, the graph is horizontally stretched by a factor of $\frac{1}{k}$. Additionally, a negative sign inside the function $f(-x)$ represents a reflection across the y-axis.

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Solution

To graph these functions, we first establish a table of values for the parent function $y=x^2$ and compare how the transformations affect the output ($y$) for given inputs ($x$).

Table of Values

Part (a) $y=(3x{)}^2$

This function represents a horizontal compression of the parent function $y=x^2$ by a factor of $\frac{1}{3}$. Because the $x$ value is multiplied by 3 before being squared, the $y$ values increase much more rapidly. For example, when $x=1$, $y=(3\cdot 1{)}^2=9$.

Part (b) $y=(-3x{)}^2$

This function involves a reflection across the y-axis and a horizontal compression. However, since the parent function $y=x^2$ is even (symmetric about the y-axis), and $(-3x{)}^2=9x^2$, the graph is identical to the graph in part (a).

Part (c) $y={\left(\frac{x}{3}\right)}^2$

This function represents a horizontal stretch of the parent function $y=x^2$ by a factor of 3. The $x$ value is divided by 3 before being squared, meaning the parabola opens much wider. For example, it takes $x=3$ to reach the height of $y=1$ (since $(\frac{3}{3}{)}^2=1$).

Part (d) $y={\left(-\frac{x}{3}\right)}^2$

Similar to part (b), the negative sign indicates a reflection across the y-axis. Because ${\left(-\frac{x}{3}\right)}^2={\left(\frac{x}{3}\right)}^2$, the resulting graph is identical to the stretched parabola in part (c).

Graphical Representation

The following image shows the comparison between the parent function and the transformed functions:

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Key Formulas or Methods Used

• Horizontal Dilation: For $y=f(kx)$, the graph is compressed horizontally if $|k|>1$ and stretched if $0<|k|<1$.
• Horizontal Reflection: $y=f(-x)$ reflects the graph across the y-axis.
• Even Function Property: For $y=x^2$, $f(x)=f(-x)$, meaning reflections across the y-axis do not change the appearance of the graph.
• Point Plotting: Calculating specific coordinates to verify the shape of the transformation.

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Summary of Steps

1. Identify the parent function: Start with $y=x^2$ and its standard points (e.g., $(0,0),(1,1),(2,4)$).
2. Apply horizontal scaling: For $y=(3x{)}^2$, multiply the $x$-coordinates of the parent function by $\frac{1}{3}$ to find the new points at the same $y$-level.
3. Apply horizontal stretching: For $y=(\frac{x}{3}{)}^2$, multiply the $x$-coordinates of the parent function by $3$.
4. Account for reflections: Recognize that because the values are squared, $(-kx{)}^2$ results in the same graph as $(kx{)}^2$.
5. Sketch and Verify: Draw the curves through the calculated points and use a graphing calculator to confirm the widths of the parabolas.