Question Statement

Draw the graphs of the 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) $y=\frac{x^2}{3}$ (d) $y=-\frac{x^2}{3}$

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Background and Explanation

To graph transformations of the parent function $y=x^2$, we apply vertical stretching, shrinking, and reflections. Multiplying the function by a constant $a$ results in a vertical stretch if $|a|>1$ and a vertical compression (shrink) if $0<|a|<1$, while a negative sign indicates a reflection across the $x$-axis.

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Solution

To graph these functions, we first calculate a table of values for $x$ ranging from $-5$ to $5$.

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

The function $y=3x^2$ is a vertical stretch of the parent function $y=x^2$ by a factor of 3. This means for every $x$ value, the $y$ value is three times larger than the original. The graph appears "narrower" or steeper than $y=x^2$.

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

The function $y=-3x^2$ involves two transformations: a vertical stretch by a factor of 3 and a reflection across the $x$-axis. The negative sign flips the parabola downward, so all $y$ values (except at the origin) become negative.

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

The function $y=\frac{x^2}{3}$ is a vertical compression (or shrink) of the parent function by a factor of $\frac{1}{3}$. Each $y$ value is one-third of the original $y$ value from $y=x^2$. This makes the parabola appear "wider."

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

The function $y=-\frac{x^2}{3}$ is a vertical compression by a factor of $\frac{1}{3}$ combined with a reflection across the $x$-axis. The graph opens downward and is wider than the parent function.

Graphical Representation

Below are the sketches of the functions based on the calculated points:

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Key Formulas or Methods Used

• Parent Function: $y=x^2$ (a parabola with vertex at $(0,0)$).
• Vertical Stretch/Compression: $y=a\cdot f(x)$. If $|a|>1$, it stretches; if $0<|a|<1$, it compresses.
• Reflection: $y=-f(x)$ reflects the graph across the $x$-axis.
• Point-by-Point Plotting: Calculating specific $(x,y)$ coordinates to determine the shape of the curve.

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Summary of Steps

1. Create a table of $x$ values (e.g., from $-5$ to $5$).
2. Calculate the $y$ values for the parent function $y=x^2$.
3. Multiply the parent $y$ values by the coefficient $a$ (3, -3, $1/3$, or $-1/3$) to find the new $y$ coordinates.
4. Plot the resulting points on a Cartesian plane.
5. Connect the points with a smooth curve to form the transformed parabolas.
6. Observe that positive coefficients open upward, negative coefficients open downward, and larger coefficients create narrower graphs.