How do you find the equation of the secant line of #f(x)=x^2-5x# through the points [1,8]?

Question

Luke Alvarez · Accepted Answer

The line will pass through the points #(1, f(1))# and #(8, f(8))# #f(1) = 1^2 - 5(1)# #f(1) = -4# #f(8) = 8^2 - 5(8)# #f(8) = 24# The line will pass through the points #(1, -4)# and #(8, 24)# The slope is #m =(24 - (-4))/(8 - 2) = 28/7 = 4# The point-slope form of the equation of a line is: #y = m(x - x_0) + y_0# Substitute #4# for m, 1 for #x_0# and -4 for #y_0# #y = 4(x - 1) - 4# #y = 4x - 4 - 4# #y = 4x - 8#

Christopher Agresta · Answer

# y = 4x-8 #

We have #f(x) = x^2 - 5x#

When #x=1 => f(x) = 1-5 = -4#
When #x=8 => f(x) = 64-40 = 24#

So the required secant line passes through the points #(1, -4)# and #(8, 24)#.

We can calculate the slope of the secant line using

#m=(Delta y)/(Delta x) = (24-(-4)) / (8-1) = 28/7 = 4#

So using the factthat the line passes through #(1,-4)# and has slope #4# and Using the formula #y - y_0 = m(x - x_0) #, the required equation is given by:

# y - (-4)=4(x-1) #
# :. y +4 = 4x-4 #
# :. y = 4x-8 #

NB: We could have equally used the other coordinate

Which we can confirm graphically:

Evelyn Agard · Answer

undefined To find the equation of the secant line of the function $ f(x) = x^2 - 5x $ through the points $[1,8]$, you can use the point-slope form of a linear equation.

1. First, find the slope of the secant line using the two given points $[1,8]$ and another point on the secant line. You can choose any point on the line, but for simplicity, let's choose a second point $ [x, f(x)] $ on the line.

2. The slope of the secant line is given by the formula:
$$ \text{Slope} = \frac{\text{change in } y}{\text{change in } x} $$

3. Plug the coordinates of the given points into the formula to find the slope:
$$ \text{Slope} = \frac{f(x) - 8}{x - 1} $$

4. Now, substitute the function $ f(x) = x^2 - 5x $ into the formula:
$$ \text{Slope} = \frac{x^2 - 5x - 8}{x - 1} $$

5. Next, simplify the expression:
$$ \text{Slope} = \frac{x^2 - 5x - 8}{x - 1} = \frac{(x - 1)(x - 8)}{x - 1} $$
$$ \text{Slope} = x - 8 $$

6. Now that you have the slope, you can use the point-slope form of the equation of a line to find the equation of the secant line:
$$ y - y_1 = m(x - x_1) $$

7. Substitute the coordinates of the given point $[1,8]$ and the slope $m = x - 8$ into the point-slope form:
$$ y - 8 = (x - 8)(x - 1) $$

8. Expand and simplify the expression:
$$ y - 8 = x^2 - 9x + 8 $$
$$ y = x^2 - 9x + 16 $$

Therefore, the equation of the secant line of $ f(x) = x^2 - 5x $ through the points $[1,8]$ is $ y = x^2 - 9x + 16 $.