Geometry - How to find the equation of a curve

Which equation has the x- and y-intercepts and ?

$y = -\frac{7}{2}x + 7$

$y = \frac{7}{2} x +3$

$y = \frac{2}{7} x + 2$

$y = -\frac{2}{7} + 3$

Explanation

Since this line has the y-intercept , we know that in the form ,

We can plug in the other intercept's coordinates for and to solve for :

subtract

divide by

$\frac{-7}{2} = m$

The line is $y = -\frac{7}{2}x + 7$

Which equation would have an x-intercept at $\small x=2$ and a y-intercept at $\small y=-3$ ?

$\small y = \frac{3}{2}x - 3$

$\small y = -\frac{3}{2}x - 3$

$y = \frac{2}{3}x - 3$

$\small y = -\frac{2}{3}x + 2$

$\small y = \frac{3}{2}x + 3$

Explanation

We're writing the equation for a line passing through the points $\small (0,-3)$ and $\small (2,0)$ . Since we already know the y-intercept, we can figure out the slope of this line and then write a slope-intercept equation.

To determine the slope, divide the change in y by the change in x:

$\small \frac{-3 - 0 }{ 0 - 2 } = \frac{-3}{-2} = \frac{3}2$

The equation for this line would be $\small y = \frac{3}{2}x - 3$ .

Which equation has a y-intercept at 2 and x-intercepts at -1 and 6?

$\small y = \frac{-1}{3}(x+1)(x-6)$

$\small y = \frac{1}{3}(x-1)(x+6)$

$\small y=3 (x+1)(x-2 )$

$\small \small y = (x+1)(x-6)$

$\small y = -3(x+1)(x-2)$

Explanation

In order for the equation to have x-intercepts at -1 and 6, it must have $\small (x+1)$ and as factors. This leaves us with only 2 choices, $\small \small y = (x+1)(x-6)$ or $\small y = \frac{-1}{3}(x+1)(x-6)$

This equation must also have a y-intercept of 2. This means that plugging in 0 for x will gives us a y-value of 2. Because we have two options, we could plug in 0 for x in each to see which gives us an answer of 2:

a) $\small y = (0+1)(0-6) = 1*-6 = -6$ we can eliminate that choice

b) $\small y= -\frac{1}{3}(0+1)(0-6) = -\frac{1}{3}*1*-6 = 2$ this must be the right choice.

If we hadn't been given multiple options, we could have set up the following equation to figure out the third factor:

$\small 2 = C(0+1)(0-6)$

$\small 2 = C*1*-6$

$\small 2 = -6C$ divide by -6

$\small -\frac{1}{3} = C$

Find the -intercept of:

Explanation

To find the x-intercept, we need to find the value of when .

So we first set to zero.

turns into

Lets subtract from both sides to move to one side of the equation.

After doing the arithmetic, we have

Divide by from both sides

$\frac{-4x}{-4}=\frac{2}{-4}$

What is the -intercept of:

Explanation

The x-intercept can be found where

turns into

Lets subtract from both sides to solve for .

After doing the arithmetic we have

Divide both sides by

$\frac{-3x}{-3}=\frac{3}{-3}$

Write the equation of a line with intercepts and

$y = \frac{1}{4}x +1$

$y = -\frac{1}{4}x + 1$

Explanation

The line will eventually be in the form where is the y-intercept.

The y-intercept in this case is .

To find the equation, plug in for , and the other point, as x and y:

add to both sides

This means the equation is

If a line's -intercept is . and the -intercept is , what is the equation of the line?

$-\frac{1}{5}x+y=-1$

$\frac{1}{5}x-\frac{1}{5}y=-1$

$-\frac{1}{5}x+y=\frac{1}{5}$

Explanation

Write the equation in slope-intercept form:

We were given the -intercept, , which means :

Given the -intercept is , the point existing on the line is . Substitute this point into the slope-intercept equation and then solve for to find the slope: