How to find x or y intercept - Geometry

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Question

True or false: The lines of the equations

and

have the same -intercept.

Answer

The -intercept of a line is the point at which it intersects the -axis; its -coordinate at this point is 0, so its -coordinate can be found by substituting 0 for in the equation and solving for . Do this with both equations:

$2 \cdot 0 +y = 12$

The -intercept of the line is at .

$\frac{2y }{2}= \frac{12}{2}$

The -intercept of the line is at .

The two lines have different -intercepts.

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Question

True or false: the lines of the equations

and

have the same -intercept.

Answer

The -intercept of a line is the point at which it intersects the -axis; its -coordinate at this point is 0.

The -coordinate of the line of the equation

can be found by substituting 0 for in the equation and solving for :

The -intercept of this line is at .

The line with equation is a horizontal line with its -intercept at , so the -intercept line of the equation has its -intercept at as well.

Both lines indeed have the same -intercept.

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Question

What is the x-intercept for the line ?

Answer

To find the x-intercept, plug in for , since the x-axis is where .

add to both sides

divide both sides by

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Question

Find the y-intercept of the line $-14x+\frac{1}{4}y=2$ .

Answer

Recall that at the y-intercept, the line crosses the y-axis. This means that at the y-intercept, the value of the x-coordinate is .

Plug in for into the equation to find the y-intercept.

$-14x+\frac{1}{4}y=2$

$-14(0)+\frac{1}{4}y=2$

$\frac{1}{4}y=2$

The y-intercept for this line is .

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Question

What are the and -intercepts of the line defined by the equation:

Answer

To find the intercepts of a line, we must set the and values equal to zero and then solve.

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Question

Given the line what is the sum of the and intercepts?

Answer

The intercepts cross an axis.

For the intercept, set to get

For the intercept, set to get

So the sum of the intercepts is .

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Question

Find the y-intercept of the line $2x+\frac{2}{3}y=6$ .

Answer

Recall that at the y-intercept, the line crosses the y-axis. This means that at the y-intercept, the value of the x-coordinate is .

Plug in for into the equation to find the y-intercept.

$2x+\frac{2}{3}y=6$

$2(0)+\frac{2}{3}y=6$

$\frac{2}{3}y=6$

The y-intercept for this line is .

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Question

What is the -intercept of the following line:

Answer

The -intercept is the point where the y-value is equal to 0. Therefore,

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Question

Which of the following statements regarding the x and y intercepts of the equation is true?

Answer

To find the x-intercept, we simply plug into our function. giving us . We can factor that equation, making it . We can not solve for , and we get . To find the y-intercept, we do the same thing, however this time, we plug in instead. This leaves us with . With an x-intercept of and a y-intercept of , it is clear that the y-intercept is greater than the x-intercept.

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Question

Find the -intercept of the following function.

Answer

To find the x-intercept, set y equal to 0.

Now solve for x by dividing by 3 on both sides.

$\frac{9}{3}=\frac{3}{3}x$

This reduces to,

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Question

Find the -intercept of the following function.

Answer

To find the y-intercept, set x equal to 0.

Now solve for y.

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Question

Which is the x-intercept for the line $\small y = 2x - 5$ ?

Answer

The x-intercept of a line is the x-value where the line hits the x-axis. This occurs when y is 0. To determine the x-value, plug in 0 for y in the original equation, then solve for x:

$\small 0 = 2x - 5$ add 5 to both sides

$\small 5 = 2x$ divide by 2

$\small 2.5 = x$

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Question

Find the x-intercept(s) for the circle $\small (x-4)^2 + y^2 = 9$

Answer

The x-intercepts of any curve are the x-values where the curve is intersecting the x-axis. This happens when y = 0. To figure out these x-values, plug in 0 for y in the original equation and solve for x:

$\small (x-4)^2 + 0^2 = 9$ adding 0 or 0 square doesn't change the value

$\small (x-4)^2 = 9$ take the square root of both sides

$\small x - 4 = \pm 3$ this means there are two different potential values for x, and we will have to solve for both. First:

add 4 to both sides

$\small x = 1$

Second: $\small x - 4 = 3$ again, add 4 to both sides

Our two answers are and .

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Question

Which is neither an x- or y-intercept for the parabola $\small y = x^2 - 16$

Answer

The y-intercept(s) occur where the graph intersects with the y-axis. This is where x=0, so we can find these y-values by plugging in 0 for x in the equation:

$\small y = 0^2 - 16$

$\small y = -16$

The x-intercept(s) occur where the graph intersects with the x-axis. This is where y=0, so we can find these x-values by plugging in 0 for y in the equation:

$\small 0 = x^2 - 16$ add 16 to both sides

$\small 16 = x^2$ take the square root

$\small \pm 4 = x$

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Question

Give the coordinate pair(s) where $\small \small (y+3)^2 - (x-2)^2 = 21$ intersects with the y-axis.

Answer

To find where the graph hits the y-axis, plug in 0 for x:

$\small (y+3)^2 - (0-2)^2 = 21$ first evaluate 0 - 2

$\small (y+3)^2 - (-2)^2 = 21$ then square -2

$\small (y+3)^2 - 4 = 21$ add 4 to both sides

$\small (y+3)^2 = 25$ take the square root of both sides

$\small y + 3 = \pm 5$ now we have 2 potential solutions and need to solve for both

a) $\small y + 3 = 5$

$\small y = 2$

b) $\small y +3 = -5$

$\small y = -8$

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Question

What is the x-intercept of the line $y = -\frac{2}{3}x + 6$

Answer

To determine the x-intercept, plug in for , since the x-axis is where .

$0 = -\frac{2}{3} x + 6$ subtract from both sides

$-6 = -\frac{2}{3} x$ multiply both sides by

divide both sides by

The x-intercept is

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Question

Find the y-intercept for the line

.

Answer

To find the y-intercept, plug in 0 for x, since the y-axis is where x = 0.

subtract 24 from both sides

divide by -3

The y-intercept is

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Question

Find the x-intercept for the line

$y = -\frac{1}{3} x + 5$ .

Answer

To find the x-intercept, plug in 0 for y, since the x-axis is where y = 0

$0 = -\frac{1}{3} x + 5$ subtract 5 from both sides

$-5 = -\frac{1}{3}x$ multiply both sides by -3

The x-intercept is

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Question

A rocket is fired that follows the below parabolic path where h(t) represents the height (in feet) over time (in seconds). How much time did the rocket spend in the air?

$h(t)=256t-16t^{2}$

Answer

To solve, you must find the zeros or x-intercepts for the function h(t).

To find x-intercepts, we must plug in 0 for h(t).

$h(t)=256t-16t^{2}$

$0=256t-16t^{2}$

Factor to solve - divide by 16t.

Set both parts equat to zero.

$t = 0\ and\ t=16$

The answer is t=16 because when t=0, that is when the rocket is first being fired from the ground, the rocket then returns to the ground after having spent 16 seconds in the air.

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Question

Find the y-intercept(s) for the circle given by the equation below:

$(x-5)^{2}+(y+2)^{2}=25$

Answer

To find the y-intercept for any equation, plug in 0 for x in the original equation.

$(0-5)^{2}+(y+2)^{2}=25$

Simplify

$25+(y+2)^{2}=25$

Subtract 25 from both sides

$(y+2)^{2}=0$

Take the square root of both sides

Subtract 2 from both sides to get the y-intercept.

This circle has one y-intercept.

It is possible for a circle to have one y-intercept, two y-intercepts or no y-intercepts.

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