Intermediate Geometry : How to find the equation of a curve

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #1 : How To Find The Equation Of A Curve

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

Possible Answers:

Correct answer:

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:

Add  to each side of the equation:

Divide each side of the equation by :

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

 

By subtracting  on both sides, we can rearrange the equation to put it into standard form:

Example Question #1 : How To Find The Equation Of A Curve

Find the -intercept of:

Possible Answers:

Correct answer:

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

Example Question #3 : How To Find The Equation Of A Curve

What is the -intercept of:

Possible Answers:

Correct answer:

Explanation:

To find the y-intercept, we set 

So

turns into

.

After doing the arithmetic we get

.

Example Question #4 : How To Find The Equation Of A Curve

What is the -intercept of:

Possible Answers:

Correct answer:

Explanation:

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 

Example Question #5 : How To Find The Equation Of A Curve

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

Possible Answers:

Correct answer:

Explanation:

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.

Example Question #6 : How To Find The Equation Of A Curve

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

Possible Answers:

Correct answer:

Explanation:

In order for the equation to have x-intercepts at -1 and 6, it must have and as factors. This leaves us with only 2 choices, or

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) we can eliminate that choice

b) 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:

divide by -6

Example Question #7 : How To Find The Equation Of A Curve

Which equation would have an x-intercept at and a y-intercept at ?

Possible Answers:

Correct answer:

Explanation:

We're writing the equation for a line passing through the points and . 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:

The equation for this line would be .

Example Question #8 : How To Find The Equation Of A Curve

Write the equation of a line with intercepts and

Possible Answers:

Correct answer:

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

Example Question #2 : How To Find The Equation Of A Curve

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

Possible Answers:

Correct answer:

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 
 
The line is
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