How to find the equation of a curve - Geometry

Card 0 of 36

Question

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

Answer

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:

Add to each side of the equation:

Divide each side of the equation by :

$m=\frac{1}{5}$

Substituting the value of back into the slope-intercept equation, we get:

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

By subtracting $-\frac{1}{5}x$ on both sides, we can rearrange the equation to put it into standard form:

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

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Question

Find the -intercept of:

Answer

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

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

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Question

What is the -intercept of:

Answer

To find the y-intercept, we set

So

turns into

.

After doing the arithmetic we get

.

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Question

What is the -intercept of:

Answer

The x-intercept can be found where

So

turns into

.

Lets subtract from both sides to solve for .

After doing the arithmetic we have

.

Divide both sides by

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

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Question

Suppose two intercepts create a line. If the -intercept is and -intercept is , what is the equation of the line?

Answer

Rewrite the intercepts in terms of points.

X-intercept of 1: .

Y-intercept of 2:

Write the slope-intercept form for linear equations.

Substititute the y-intercept into the slope-intercept equation.

Substitute both the x-intercept point and the y-intercept into the equation to solve for slope.

Rewrite by substituting the values of and into the y-intercept form.

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Question

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

Answer

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:

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

$\small 2 = C1-6$

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

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

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Question

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

Answer

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

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Question

Write the equation of a line with intercepts and

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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:

Add to each side of the equation:

Divide each side of the equation by :

$m=\frac{1}{5}$

Substituting the value of back into the slope-intercept equation, we get:

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

By subtracting $-\frac{1}{5}x$ on both sides, we can rearrange the equation to put it into standard form:

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

Compare your answer with the correct one above

Question

Find the -intercept of:

Answer

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

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

Compare your answer with the correct one above

Question

What is the -intercept of:

Answer

To find the y-intercept, we set

So

turns into

.

After doing the arithmetic we get

.

Compare your answer with the correct one above

Question

What is the -intercept of:

Answer

The x-intercept can be found where

So

turns into

.

Lets subtract from both sides to solve for .

After doing the arithmetic we have

.

Divide both sides by

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

Compare your answer with the correct one above

Question

Suppose two intercepts create a line. If the -intercept is and -intercept is , what is the equation of the line?

Answer

Rewrite the intercepts in terms of points.

X-intercept of 1: .

Y-intercept of 2:

Write the slope-intercept form for linear equations.

Substititute the y-intercept into the slope-intercept equation.

Substitute both the x-intercept point and the y-intercept into the equation to solve for slope.

Rewrite by substituting the values of and into the y-intercept form.

Compare your answer with the correct one above

Question

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

Answer

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:

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

$\small 2 = C1-6$

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

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

Compare your answer with the correct one above

Question

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

Answer

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

Compare your answer with the correct one above

Question

Write the equation of a line with intercepts and

Answer

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

Compare your answer with the correct one above

Question

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

Answer

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

Compare your answer with the correct one above

Question

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

Answer

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:

Add to each side of the equation:

Divide each side of the equation by :

$m=\frac{1}{5}$

Substituting the value of back into the slope-intercept equation, we get:

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

By subtracting $-\frac{1}{5}x$ on both sides, we can rearrange the equation to put it into standard form:

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

Compare your answer with the correct one above

Question

Find the -intercept of:

Answer

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

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

Compare your answer with the correct one above

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