Write a linear function \( f \) that has the indicated values. Sketch the graph of \( f \). \( f(-5)=-15, f(2)=13 \) \( f(x)=\square \) (Simplify your answer. Type an expression using \( x \) as the variable. Type your answer in slope-intercept form.) Use the graphing tool to graph the function. Click to enlarge graph

To find the linear function \( f(x) \), we first need to determine the slope \( m \) using the two given points: \( (-5, -15) \) and \( (2, 13) \). The formula for the slope is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{13 - (-15)}{2 - (-5)} = \frac{13 + 15}{2 + 5} = \frac{28}{7} = 4. \] Next, we can use point-slope form to find the equation of the line. We will use point \( (2, 13) \): \[ f(x) - 13 = 4(x - 2). \] Simplifying this gives: \[ f(x) - 13 = 4x - 8 \implies f(x) = 4x + 5. \] So, the linear function is: \[ f(x) = 4x + 5. \] Now, for the sketch of the graph, you would plot points using the equation \( f(x) = 4x + 5 \). The y-intercept (where \( x=0 \)) is \( 5 \), and you can use the slope of \( 4 \) to plot additional points. For example, if you go one unit to the right (to \( x=1 \)), you go up \( 4 \) units, reaching \( (1, 9) \). Plotting these points and drawing a straight line through them will give you the graph of the function. Now, let's explore some interesting historical background and real-world applications related to linear functions! Linear functions have been around since ancient civilizations used them to solve practical problems such as land measurement. The ancient Greeks, particularly Euclid, laid the groundwork for algebra and geometric principles, allowing mathematicians to model relationships between quantities—essentially the first steps towards understanding linear equations. In the real world, linear functions are exceptionally useful! Engineers, economists, and data analysts regularly use them to model relationships and trends. For instance, you could use a linear function to forecast sales based on historical data, or an engineer might apply it to budget project costs based on the price per unit. These functions simplify complex relationships, making predictions just a bit easier!