Find the equation of the secant line to the graph of y= 2x^3-3x^2+4x-1 from x=1 to x=3. Give your answer in slope-intercept form.
To find the equation of the secant line, we need to find the slope and y-intercept.
Step 1: Find the coordinates of two points on the graph.
Plug x=1 and x=3 into the equation y=2x^3-3x^2+4x-1 to find their corresponding y-values.
When x=1:
y = 2(1)^3 - 3(1)^2 + 4(1) - 1
= 2 - 3 + 4 - 1
= 2
So, the first point is (1, 2).
When x=3:
y = 2(3)^3 - 3(3)^2 + 4(3) - 1
= 54 - 27 + 12 - 1
= 38
So, the second point is (3, 38).
Step 2: Find the slope.
The slope of the secant line is given by the formula:
slope = (change in y) / (change in x)
(change in y) = y2 - y1 = 38 - 2 = 36
(change in x) = x2 - x1 = 3 - 1 = 2
slope = 36 / 2 = 18
Step 3: Find the y-intercept.
We can use the point-slope form of a linear equation to find the y-intercept. The formula is:
y - y1 = m(x - x1)
Using the point (1, 2) and slope m = 18:
y - 2 = 18(x - 1)
Simplify:
y - 2 = 18x - 18
y = 18x - 16
So, the equation of the secant line in slope-intercept form is y = 18x - 16.