Question Statement

Draw the graphs of the given functions and then sketch the graphs of other functions using translation. Verify the results using a graphical calculator:

Base function: $y=x$

(a) $y=x+5$ (b) $y=x-5$ (c) $y=5x$ (d) $y=-5x$

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Background and Explanation

This problem explores function transformations of the linear parent function $f(x)=x$. Adding or subtracting a constant results in a vertical translation, while multiplying the function by a constant results in a vertical stretch and/or reflection.

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Solution

To graph these functions, we first calculate a table of values for $x$ ranging from $-3$ to $3$.

Part (a) $y=x+5$

This function represents a vertical translation of the parent function $y=x$. Since we are adding $5$ to the output, every point on the graph of $y=x$ moves upward by 5 units. For example, the origin $(0,0)$ moves to $(0,5)$.

Part (b) $y=x-5$

This function represents a vertical translation of the parent function $y=x$. Since we are subtracting $5$ from the output, every point on the graph of $y=x$ moves downward by 5 units. The origin $(0,0)$ moves to $(0,-5)$.

Part (c) $y=5x$

This function represents a vertical stretch by a factor of $5$. The slope of the line increases from $1$ to $5$, making the graph much steeper. While the $y$-intercept remains at $(0,0)$, all other $y$-values are multiplied by $5$.

Part (d) $y=-5x$

This function involves two transformations: a vertical stretch by a factor of $5$ and a reflection across the $x$-axis (due to the negative sign). The resulting line has a steep negative slope, passing through the origin $(0,0)$ but decreasing as $x$ increases.

Graphical Representation

Below are the visual representations of these transformations:

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Key Formulas or Methods Used

• Vertical Translation: $y=f(x)+k$ shifts the graph up if $k>0$ and down if $k<0$.
• Vertical Stretch: $y=a\cdot f(x)$ stretches the graph vertically if $|a|>1$.
• Reflection: $y=-f(x)$ reflects the graph across the $x$-axis.
• Table of Values: Evaluating the function at specific $x$ points to determine coordinates.

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Summary of Steps

1. Create a table of values for the parent function $y=x$.
2. Calculate the new $y$-values for each transformation by applying the arithmetic change (adding, subtracting, or multiplying).
3. Identify the type of transformation: parts (a) and (b) are shifts, while (c) and (d) are stretches and reflections.
4. Plot the points on a Cartesian plane and connect them to form straight lines.
5. Verify the steepness and intercepts using a graphical calculator.