Subjects algebra

Linear Transformations

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1. **State the problem:** We start with the line $y = x$ and transform it into $y = \frac{x}{4} + 2$. We need to find the slope and describe how the line is shifted. 2. **Recall the slope-intercept form:** The equation of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. 3. **Identify the slope and y-intercept of the original line:** For $y = x$, the slope $m = 1$ and the y-intercept $b = 0$. 4. **Identify the slope and y-intercept of the transformed line:** For $y = \frac{x}{4} + 2$, the slope $m = \frac{1}{4}$ and the y-intercept $b = 2$. 5. **Compare slopes:** The new slope $\frac{1}{4}$ is less than the original slope $1$, so the line is **flatter**. 6. **Describe the shift:** The y-intercept changed from $0$ to $2$, so the line is shifted **upwards by 2 units**. **Final answer:** The slope is $\frac{1}{4}$, and the line is shifted upwards by 2 units, making it flatter than the original line.