X-intercept and y-intercept - SAT Math

Card 0 of 148

Question

Find the y-intercept of the following line.

Answer

To find the y-intercept of any line, we must get the equation into the form

where m is the slope and b is the y-intercept.

To manipulate our equation into this form, we must solve for y. First, we must move the x term to the right side of our equation by subtracting it from both sides.

To isolate y, we now must divide each side by 3.

$\frac{3y}{3}=\frac{-4x}{3}-\frac{12}{3}$

$y=\frac{-4}{3}x-4$

Now that our equation is in the desired form, our y-intercept is simply

$\small b=-4$

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Question

What is the -intercept of the following equation?

Answer

The easiest way to solve for this kind of simple -intercept is to set equal to . You can then solve for the value in order to find the relevant intercept.

Solve for :

Divide both sides by 40:

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Question

Solve for the -intercepts of this equation:

Round each of your answers to the nearest tenth.

Answer

For an equation like this, you should use the quadratic formula to solve for the roots. We can easily get our equation into proper form by substituting for :

Recall that the general form of the quadratic formula is:

$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Based on our equations, the following are your formula values:

Therefore, the quadratic formula will be:

$\frac{-13\pm\sqrt{13^2-4(5)(-178)}}{2*5}$

Simplifying, you get:

$\frac{-13\pm\sqrt{169+3560}}{10}$

$\frac{-13\pm\sqrt{3729}}{10}$

Using a calculator, you will get:

and

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Question

Solve for the -intercepts of this equation:

Round each of your answers to the nearest tenth.

Answer

For an equation like this, you should use the quadratic formula to solve for the roots. We can easily get our equation into proper form by substituting for :

Recall that the general form of the quadratic formula is:

$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Based on our equations, the following are your formula values:

Therefore, the quadratic formula will be:

$\frac{-6\pm\sqrt{(6)^2-4(2)(-15)}}{2(2)}$

Simplifying, you get:

$\frac{-6\pm\sqrt{36+120}}{4}$

$\frac{-6\pm\sqrt{156}}{4}$

Using a calculator, you will get:

and

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Question

Solve for the -intercepts of this equation:

$\frac{3x+4+y}{x-2} = 2x+5$

Round each of your answers to the nearest tenth.

Answer

For an equation like this, you should use the quadratic formula to solve for the roots. We can easily get our equation into proper form by substituting for . Then, we need to get it into standard form:

$\frac{3x+4+0}{x-2} = 2x+5$

Recall that the general form of the quadratic formula is:

$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Based on our equations, the following are your formula values:

Therefore, the quadratic formula will be:

$\frac{2\pm\sqrt{(-2)^2-4(2)(-14)}}{2(2)}$

Simplifying, you get:

$\frac{2\pm\sqrt{4+112}}{4}$

$\frac{2\pm\sqrt{116}}{4}$

Using a calculator, you will get:

and

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Question

What are the -intercepts of the following equation?

Round each of your answers to the nearest tenth.

Answer

There are two ways to solve this. First, you could substitute in for :

Take the square-root of both sides and get:

$y-5 = \pm11$

$y=5\pm11$

Therefore, your two answers are and .

You also could have done this by noticing that the problem is a circle of radius , shifted upward by .

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Question

Find the -intercepts of the following equation:

Round each of your answers to the nearest tenth.

Answer

There are two ways to solve this. First, you could substitute in for :

Take the square-root of both sides and get:

$y+14 = \pm5$

$y= -14\pm5$

Therefore, your two answers are and .

You also could have done this by noticing that the problem is a circle of radius , shifted downward by .

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Question

Find the -intercepts of the following equation:

Round each of your answers to the nearest tenth.

Answer

For an equation like this, you should use the quadratic formula to solve for the roots. We can easily get our equation into proper form by substituting for . Then, we need to get it into standard form:

Recall that the general form of the quadratic formula is:

$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Based on our equations, the following are your formula values:

Therefore, the quadratic formula will be:

$\frac{-5\pm\sqrt{5^2-4(1)(2)}}{2(2)}$

Simplifying, you get:

$\frac{-5\pm\sqrt{25-8}}{4}$

$\frac{-5\pm\sqrt{17}}{4}$

Using a calculator, you will get:

and

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Question

What is the x-intercept of the above equation?

Answer

To find the x-intercept, you must plug in for .

This gives you,

and you must solve for .

First, add to both sides which gives you,

.

Then divide both sides by to get,

.

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Question

What is the x-intercept of the given equation?

Answer

In order to determine the x-intercept, we will need to let , and solve for .

Divide both sides by two.

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

The answer is: $\frac{1}{2}$

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Question

What is the y-intercept of the function? $y=\frac{2}{x-1}+\frac{3}{x+1}$

Answer

The y-intercept is the value of when .

Substitute zero into the x-variable in the equation.

$y=\frac{2}{0-1}+\frac{3}{0+1} = -2+3 = 1$

The y-intercept is .

The answer is:

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Question

What is the x-intercept and y-intercept of:

Answer

The x-intercept is the value of x that will result in the y value equalling zero and the y-intercept is the y value that results when the x equals zero.

Therefore we replace x with zero to find the y-intecerpt.

Now we replace the y with a zero and find our x-intercept

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Question

What is the y-intercept and x-intercept of the function:

Answer

To find the x and y intercept of the function we need to do algebraic opperations first and then plug in zero for x to find the y-intercept and plug in zero for y to find the x-intercept.

y-intercept

x-intercept

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Question

Define $f(x) = 2 \cdot 3^{x-4}- 7$

Give the -coordinate of the -intercept of the graph of (nearest hundredth).

Answer

Set and solve for :

$f(x) = 2 \cdot 3^{x-4}- 7$

$2 \cdot 3^{x-4}- 7= 0$

$2 \cdot 3^{x-4}=7$

$3^{x-4}=\frac{7}{2} = 3.5$

$\ln \left (3^{x-4} \right )= \ln 3.5$

$(x-4 ) \ln 3 = \ln 3.5$

$x-4 = \frac{\ln 3.5}{ \ln 3} \approx \frac{1.2528}{1.0986} \approx 1.14$

$x\approx 5.14$

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Question

$h(x)=6- \log _{3}\left ( x-7 \right )$

Give the -coordinate of the -intercept of the graph of (nearest hundredth).

Answer

Set and solve for :

$6- \log _{3}\left ( x-7 \right )=0$

$\log _{3}\left ( x-7 \right )=6$

$x-7 = 3^{6}$

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Question

Define $f(x) = 2 \cdot 3^{x-4}- 7$

Give the -coordinate of the -intercept of the graph of (nearest hundredth).

Answer

Evaluate :

$f(x) = 2 \cdot 3^{x-4}- 7$

$f(0) = 2 \cdot 3^{0-4}- 7$

$= 2 \cdot 3^{-4}- 7$

$= 2 \cdot \frac{1}{81}- 7$

$= \frac{2}{81}- 7$

$=-6 \frac{79}{81} \approx -6.98$

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Question

Define $g(x)=5- \log _{4}\left ( x+5 \right )$ .

Give the -coordinate of the -intercept of the graph of (nearest hundredth).

Answer

Evaluate :

$g(x)=5- \log _{4}\left ( x+5 \right )$

$g(0)=5- \log _{4}\left ( 0+5 \right )$

$=5- \log _{4} 5$

$=5- \frac{\ln 5}{\ln 4}$

$\approx 5- \frac{1.6094}{1.3863}$

$\approx 5- 1.16$

$\approx 3.84$

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Question

Define $g(x)=5- \log _{4}\left ( x+5 \right )$ .

Give the -coordinate of the -intercept of the graph of .

Answer

Set and solve for :

$5- \log _{4}\left ( x+5 \right ) = 0$

$\log _{4}\left ( x+5 \right ) = 5$

$x+5 = 4^{5}$

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Question

Define $h(x)=6- \log _{3}\left ( x-7 \right )$ .

Give the -coordinate of the -intercept of the graph of (nearest hundredth).

Answer

Evaluate :

$h(x)=6- \log _{3}\left ( x-7 \right )$

$h(0)=6- \log _{3}\left ( 0-7 \right )$

$h(0)=6- \log _{3}\left ( -7 \right )$

A negative number cannot have a logarithm, so is an undefined expression. Therefore, the graph of has no -intercept.

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Question

Define $g(x) = 3 + \sqrt[4]{3x-5}$ .

Give the -coordinate of the -intercept of the graph of (nearest hundredth).

Answer

Evaluate :

$g(x) = 3 + \sqrt[4]{3x-5}$

$g(0) = 3 + \sqrt[4]{3\cdot 0-5}$

$g(0) = 3 + \sqrt[4]{ -5}$

A negative number does not have a real even-numbered root, so is not a real number. Therefore, the graph of has no -intercept.

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