Question

Consider the function f(x) = 4x−x² and the point P(1,3) on the graph of f.

Graph f and the secant lines passing through P(1,3) and Q(x,f(x)) for the x values of 0, 0.5, and 1.5

Answer 1

To graph the function f(x) = 4x−x² and find the secant lines, calculate the y-values for given x-values within the range. Use the slope formula and slope-intercept form of a line to find the equations of the secant lines.

Step-by-step explanation:

To graph the function f(x) = 4x−x², we can first find the y-values for different x-values within the given range. We can then plot these points on a coordinate plane to create the graph. For the given x-values of 0, 0.5, and 1.5, we can calculate the corresponding y-values as follows:

For x = 0: f(0) = 4(0) - (0)² = 0

For x = 0.5: f(0.5) = 4(0.5) - (0.5)² = 1.75

For x = 1.5: f(1.5) = 4(1.5) - (1.5)² = 3.75

Using these points, we can plot the graph of f(x) by connecting the dots. To find the secant lines passing through the point P(1,3) and each of the given x-values, we can calculate the slope of each line using the formula: slope = (change in y) / (change in x). Finally, we can use the slope-intercept form of a line (y = mx + b) to write the equations of the secant lines.