How do you find the equation of the secant line through the points where X has the given values: f(x) = x^2 + 2x; x=3, x=5?

Question

How do you find the equation of the secant line through the points where X has the given values: f(x) = x^2 + 2x;  x=3, x=5?

Ezekiel Alexander · Accepted Answer

A secant line is simply a linear equation and with two given points you can find the equation.

On the secant line, there are two points: #x = 3 and y = 3^2+2(3)= 15#; coordinate ( 3, 15 ) #x = 5 and y = 5^2+2(5)= #; coordinate ( 5, 35 ) slope of secant line =#=(35-15)/(5-3)=10# Next, find the y-intercept by solving: #y=mx + b# #15 = (10)(3)+b# #b=-15# secant line equation : #y=10x-15# I hope that was useful.

Emma Aldridge · Answer

undefined To find the equation of the secant line through the points where $ x $ has the given values $ x = 3 $ and $ x = 5 $ for the function $ f(x) = x^2 + 2x $:

1. Find the corresponding $ y $-values for the given $ x $-values:
   - When $ x = 3 $, $ f(3) = 3^2 + 2(3) = 9 + 6 = 15 $.
   - When $ x = 5 $, $ f(5) = 5^2 + 2(5) = 25 + 10 = 35 $.

2. Use the point-slope formula to find the equation of the secant line:
   $$ \text{Point-slope formula: } y - y_1 = m(x - x_1) $$
   - Let $ (x_1, y_1) $ be either of the given points $ (3, 15) $ or $ (5, 35) $, and $ (x, y) $ be any point on the secant line.
   - Find the slope $ m $ using the two points:
     $$ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} = \frac{{35 - 15}}{{5 - 3}} = \frac{{20}}{{2}} = 10 $$ 
   - Choose one of the given points and substitute its coordinates and the slope into the point-slope formula.

3. Substitute the values into the point-slope formula and simplify to get the equation of the secant line. Let's choose the point $ (3, 15) $:
   $$ y - 15 = 10(x - 3) $$
   $$ y - 15 = 10x - 30 $$
   $$ y = 10x - 30 + 15 $$
   $$ y = 10x - 15 $$

Thus, the equation of the secant line through the points $ x = 3 $ and $ x = 5 $ for the function $ f(x) = x^2 + 2x $ is $ y = 10x - 15 $.

Elijah Abram · Answer

undefined To find the equation of the secant line through the points where x has the given values, first find the corresponding y-values by substituting the given x-values into the function $ f(x) = x^2 + 2x $.

For $ x = 3 $:
$ f(3) = (3)^2 + 2(3) = 9 + 6 = 15 $

For $ x = 5 $:
$ f(5) = (5)^2 + 2(5) = 25 + 10 = 35 $

Now, you have two points: (3, 15) and (5, 35).

Next, calculate the slope of the secant line using the formula:
$$ \text{Slope} = \frac{\text{change in } y}{\text{change in } x} = \frac{35 - 15}{5 - 3} = \frac{20}{2} = 10 $$

Now that you have the slope and one point, you can use the point-slope form of a line to find the equation of the secant line:
$$ y - y_1 = m(x - x_1) $$
$$ y - 15 = 10(x - 3) $$

Simplify:
$$ y - 15 = 10x - 30 $$

Finally, rearrange to obtain the equation in slope-intercept form:
$$ y = 10x - 15 $$

So, the equation of the secant line through the points where x has the given values is $ y = 10x - 15 $.