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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:

-5x+y=-1

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

5x-y=-1

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

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

Correct answer:

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

Explanation:

Write the equation in slope-intercept form:

y=mx+b

We were given the -intercept, , which means b=-1 :

y=mx-1

Given the -intercept is , the point existing on the line is (5,0) . Substitute this point into the slope-intercept equation and then solve for to find the slope:

0=m(5)-1

Add to each side of the equation:

1=5m

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

Find the -intercept of:

y=4x+2

Possible Answers:

x=1

x=-2

x=0.5

x=2

x=-0.5

Correct answer:

x=-0.5

Explanation:

To find the x-intercept, we need to find the value of when y=0 .

So we first set to zero.

y=4x+2

turns into

0=4x+2

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

-4x=4x-4x+2

After doing the arithmetic, we have

-4x=2 .

Divide by from both sides

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

x=-0.5

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

What is the -intercept of:

y=x^2+x+9

Possible Answers:

y=10

x=9

x=2

y=-9

y=9

Correct answer:

y=9

Explanation:

To find the y-intercept, we set x=0

y=x^2+x+9

turns into

y=0^2+0+9 .

After doing the arithmetic we get

y=9 .

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

What is the -intercept of:

y=3x+3

Possible Answers:

x=-1

x=1

x=0

y=-1

y=1

Correct answer:

x=-1

Explanation:

The x-intercept can be found where y=0

y=3x+3

turns into

0=3x+3 .

Lets subtract from both sides to solve for .

-3x=3x-3x+3

After doing the arithmetic we have

-3x=3 .

Divide both sides by

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

x=-1

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Example Question #4 : 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:

y=-2x+2

x-2y=0

$y=-\frac{1}{2}x+2$

y=2x+2

$y=\frac{1}{2}x+2$

Correct answer:

y=-2x+2

Explanation:

Rewrite the intercepts in terms of points.

X-intercept of 1: (1,0) .

Y-intercept of 2: (0,2)

Write the slope-intercept form for linear equations.

y=mx+b

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

2=m(0)+b

b=2

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

0=m(1)+2

-2=m

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

y=-2x+2

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Example Question #1 : 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:

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

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

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

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

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

Correct answer:

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

Explanation:

In order for the equation to have x-intercepts at -1 and 6, it must have $\small (x+1)$ and (x-6) 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

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

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

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

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

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

Write the equation of a line with intercepts (0, 4) and (-1, 0 )

Possible Answers:

y = 4x + 4

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

y = 5x-1

y = -x+5

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

Correct answer:

y = 4x + 4

Explanation:

The line will eventually be in the form y = mx+b where is the y-intercept.

The y-intercept in this case is .

To find the equation, plug in for , and the other point, (-1,0) as x and y:

0 = m(-1)+ 4

0 = -m + 4 add to both sides

m = 4

This means the equation is y = 4x + 4

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

Which equation has the x- and y-intercepts (2,0) and (0,7) ?

Possible Answers:

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

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

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

y = 5x + 1

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

Correct answer:

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

Explanation:

Since this line has the y-intercept (0,7)

, we know that in the form y = mx+b

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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Intermediate Geometry : How to find the equation of a curve

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

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

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

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

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

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

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

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

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

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

All Intermediate Geometry Resources

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