Graph the function. \( f(x)=\frac{1}{2} x+7 \)

Certainly! Let's graph the linear function: \[ f(x) = \frac{1}{2}x + 7 \] ### 1. Understanding the Function This is a **linear function** of the form \( f(x) = mx + b \), where: - **Slope (\( m \))**: Determines the steepness and direction of the line. - **Y-intercept (\( b \))**: The point where the line crosses the y-axis. For \( f(x) = \frac{1}{2}x + 7 \): - **Slope (\( m \))**: \( \frac{1}{2} \) (The line rises 1 unit for every 2 units it moves to the right.) - **Y-intercept (\( b \))**: \( 7 \) (The line crosses the y-axis at (0, 7)). ### 2. Creating a Table of Values Choose several values for \( x \) to compute corresponding \( f(x) \): | \( x \) | \( f(x) = \frac{1}{2}x + 7 \) | |---------|-------------------------------| | -4 | \( \frac{1}{2}(-4) + 7 = -2 + 7 = 5 \) | | -2 | \( \frac{1}{2}(-2) + 7 = -1 + 7 = 6 \) | | 0 | \( \frac{1}{2}(0) + 7 = 0 + 7 = 7 \) | | 2 | \( \frac{1}{2}(2) + 7 = 1 + 7 = 8 \) | | 4 | \( \frac{1}{2}(4) + 7 = 2 + 7 = 9 \) | ### 3. Plotting the Points Plot the points from the table on a coordinate plane: - (-4, 5) - (-2, 6) - (0, 7) - (2, 8) - (4, 9) ### 4. Drawing the Line Since it's a linear function, connect the points with a straight line. The slope \( \frac{1}{2} \) means the line rises 1 unit for every 2 units it moves to the right. ### 5. ASCII Representation of the Graph While an exact graphical representation isn't possible in text, here's a simple ASCII sketch to illustrate the concept: ``` y | 10| 9| * (4,9) 8| * (2,8) 7| * (0,7) 6| * (-2,6) 5| * (-4,5) 4| 3| 2| 1| 0|____________________________ x -5 -4 -3 -2 -1 0 1 2 3 4 5 ``` ### 6. Description of the Graph - **Y-Intercept**: The line crosses the y-axis at (0, 7). - **Slope**: With a slope of \( \frac{1}{2} \), the line ascends gently. - **Direction**: Since the slope is positive, the line moves upward from left to right. ### 7. Additional Features - **X-Intercept**: To find where the line crosses the x-axis (\( f(x) = 0 \)): \[ 0 = \frac{1}{2}x + 7 \\ \frac{1}{2}x = -7 \\ x = -14 \] So, the x-intercept is at (-14, 0). You can plot this point as well to extend the line further. ### 8. Visual Tools For a more precise and visual graph, you can use graphing tools such as: - **Desmos**: [desmos.com](https://www.desmos.com/) - **GeoGebra**: [geogebra.org](https://www.geogebra.org/) - **Graphing Calculators**: Physical or online calculators can plot the function. ### 9. Summary The graph of \( f(x) = \frac{1}{2}x + 7 \) is a straight line with: - A **positive slope** (\( \frac{1}{2} \)) indicating it rises as \( x \) increases. - A **y-intercept** at (0, 7). - An **x-intercept** at (-14, 0). By plotting the key points and connecting them, you obtain the graph of the function.