Use Similar Triangles to Show Equal Slopes: CCSS.Math.Content.8.EE.B.6 - 8th Grade Math

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Question

What is the -intercept of the graph of the function

$f(x) = x^{2} + 3x - 10?$

Answer

The -intercept of the graph of a function is the point at which it intersects the -axis - that is, at which . This point is $\left (0, f(0) \right )$ , so evaluate :

$f(x) = x^{2} + 3x - 10$

$f( 0) = 0^{2} + 3 \cdot 0 - 10 = 0 + 0 - 10 = -10$

The -intercept is $\left (0, -10 \right )$ .

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Question

What is the -intercept of the graph of the function $f \left ( x\right ) = 2x^{2} - 7x + 5$ ?

Answer

The -intercept of the graph of a function is the point at which it intersects the -axis - that is, at which . This point is $\left (0, f(0) \right )$ , so evaluate :

$f \left ( x\right ) = 2x^{2} - 7x + 5$

$f \left (0\right ) = 2 \cdot 0^{2} - 7 \cdot 0 + 5$

$f \left (0\right ) = 0- 0 + 5 = 5$

The -intercept is $\left (0,5 \right )$ .

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Question

Give the -intercept, if there is one, of the graph of the equation

$y = 3(x-2)^{2} + 4 (x-7)$ .

Answer

The -intercept is the point at which the graph crosses the -axis; at this point, the -coordinate is 0, so substitute for in the equation:

$y = 3(x-2)^{2} + 4 (x-7)$

$y = 3(0-2)^{2} + 4 (0-7)$

$y = 3( -2)^{2} + 4 ( -7)$

$y = 3 \cdot 4 + 4 ( -7)$

The -intercept is the point .

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Question

Give the -intercept, if there is one, of the graph of the equation

$y = \frac{1}{x^{2}+ x} + 3$

Answer

The -intercept is the point at which the graph crosses the -axis; at this point, the -coordinate is 0, so substitute for in the equation:

$y = \frac{1}{x^{2}+ x} + 3$

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

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

However, since this expression has 0 in a denominator, it is of undefined value. This means that there is no value of paired with -coordinate 0, and, subsequently, the graph of the equation has no -intercept.

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Question

Give the -intercept, if there is one, of the graph of the equation

$y = \frac{3}{\left | 3x-8\right |}$

Answer

The -intercept is the point at which the graph crosses the -axis; at this point, the -coordinate is 0, so substitute for in the equation:

$y = \frac{3}{\left | 3x-8\right |}$

$y = \frac{3}{\left | 3 \cdot 0 -8\right |}$

$y = \frac{3}{\left | 0 -8\right |}$

$y = \frac{3}{\left | -8\right |}$

$y = \frac{3}{ 8}$

The -intercept is $\left ( 0, \frac{3}{8}\right )$ .

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Question

Give the -intercept of the line with slope $\frac{2}{5}$ that passes through point .

Answer

By the point-slope formula, this line has the equation

$y - y_{1} = m(x -x_{1})$

where

$x_{1} = 5, y_{1} = 9 ,m = \frac{2}{5}$

By substitution, the equation becomes

$y -9 = \frac{2}{5}(x -5)$

To find the -intercept, substitute 0 for and solve for :

$y -9 = \frac{2}{5}(0 -5)$

$y -9 = \frac{2}{5}( -5)$

The -intercept is the point .

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Question

Give the -intercept of the line with slope $\frac{2}{5}$ that passes through point .

Answer

By the point-slope formula, this line has the equation

$y - y_{1} = m(x -x_{1})$

where

$x_{1} = 5, y_{1} = 9 ,m = \frac{2}{5}$

By substitution, the equation becomes

$y -9 = \frac{2}{5}(x -5)$

To find the -intercept, substitute 0 for and solve for :

$0-9 = \frac{2}{5}(x -5)$

$-9 = \frac{2}{5}x -2$

$-9 +2= \frac{2}{5}x -2+2$

$-7 = \frac{2}{5}x$

$\frac{5}{2} \cdot \left (-7 \right ) = \frac{5}{2} \cdot \frac{2}{5}x$

$x = -\frac{35}{2} = -17 \frac{1}{2}$

The -intercept is $\left ( -17 \frac{1}{2}, 0 \right )$ .

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Question

Give the -intercept of the line that passes through points and .

Answer

First, find the slope of the line, using the slope formula

$m = \frac{y_{2} -y_{1}}{x_{2}-x_{1}}$

setting $x_{1} = -3, y_{1} = 4, x_{2} = 2, y_{2} = -3$ :

$m = \frac{-3 -4}{2-(-3)} = \frac{-7}{5} =- \frac{7}{5}$

By the point-slope formula, this line has the equation

$y - y_{1} = m(x -x_{1})$

where

$x_{1} = -3, y_{1} = 4, m = - \frac{7}{5}$ ; the line becomes

$y - 4 = - \frac{7}{5}(x -(-3))$

or

$y - 4 = - \frac{7}{5}(x +3)$

To find the -intercept, substitute 0 for and solve for :

$y - 4 = - \frac{7}{5}(0+3)$

$y - 4 = - \frac{7}{5}(3)$

$y - 4 = - \frac{21}{5}$

$y - 4+ 4 = - \frac{21}{5} + 4$

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

The -intercept is $\left (0, - \frac{1}{5} \right )$ .

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Question

Give the -intercept of the line that passes through points and .

Answer

First, find the slope of the line, using the slope formula

$m = \frac{y_{2} -y_{1}}{x_{2}-x_{1}}$

setting $x_{1} = -3, y_{1} = 4, x_{2} = 2, y_{2} = -3$ :

$m = \frac{-3 -4}{2-(-3)} = \frac{-7}{5} =- \frac{7}{5}$

By the point-slope formula, this line has the equation

$y - y_{1} = m(x -x_{1})$

where

$x_{1} = -3, y_{1} = 4, m = - \frac{7}{5}$ ; the line becomes

$y - 4 = - \frac{7}{5}(x -(-3))$

or

$y - 4 = - \frac{7}{5}(x +3)$

To find the -intercept, substitute 0 for and solve for :

$0 - 4 = - \frac{7}{5}(x +3)$

$- 4 = - \frac{7}{5} x - \frac{21}{5}$

$- 4+ \frac{21}{5} = - \frac{7}{5} x - \frac{21}{5}+ \frac{21}{5}$

$\frac{1}{5} = - \frac{7}{5} x$

$- \frac{5}{7} \cdot \frac{1}{5} = - \frac{5}{7}\left ( - \frac{7}{5} x \right )$

$- \frac{1}{7} =x$

The -intercept is $\left (- \frac{1}{7} , 0 \right )$ .

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Question

A line passes through and is parallel to the line of the equation . Give the -intercept of this line.

Answer

First, find the slope of the second line by solving for as follows:

$\frac{4y }{4}=\frac{ 3x-17 }{4}$

$y = \frac{3}{4}x- \frac{17}{4}$

The equation is now in the slope-intercept form ; the slope of the second line is the -coefficient $m= \frac{3}{4}$ .

The first line, being parallel to the second, has the same slope.

Therefore, we are looking for a line through with slope $m= \frac{3}{4}$ . Using point-slope form

$y - y_{1} = m(x -x_{1})$

with

$x _{1 } =5, y _{1 } = 2,m= \frac{3}{4}$ ,

the equation becomes

$y - 2 = \frac{3}{4}(x -5)$ .

To find the -intercept, substitute 0 for and solve for :

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

$- 2 = \frac{3}{4} x - \frac{15}{4}$

$- 2 + \frac{15}{4}= \frac{3}{4} x - \frac{15}{4} + \frac{15}{4}$

$\frac{3}{4} x = \frac{7}{4}$

$\frac{4} {3}\cdot \frac{3}{4} x =\frac{4} {3}\cdot \frac{7}{4}$

$x = \frac{7}{3} = 2\frac{1}{3}$

The -intercept is the point $\left (2 \frac{1}{3}, 0 \right )$ .

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Question

A line passes through and is perpendicular to the line of the equation . Give the -intercept of this line.

Answer

First, find the slope of the second line by solving for as follows:

$\frac{4y }{4}=\frac{ 3x-17 }{4}$

$y = \frac{3}{4}x- \frac{17}{4}$

The equation is now in the slope-intercept form ; the slope of the second line is the -coefficient $m= \frac{3}{4}$ .

The first line, being perpendicular to the second, has as its slope the opposite of the reciprocal of $\frac{3}{4}$ , which is $m=- \frac{4}{3}$ .

Therefore, we are looking for a line through with slope $m=- \frac{4}{3}$ . Using point-slope form

$y - y_{1} = m(x -x_{1})$

with

$x _{1 } =5, y _{1 } = 2,m=- \frac{4}{3}$ ,

the equation becomes

$y - 2 =- \frac{4}{3}(x -5)$ .

To find the -intercept, substitute 0 for and solve for :

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

$- 2 =- \frac{4}{3}x+ \frac{20}{3}$

$- 2- \frac{20}{3} =- \frac{4}{3}x+ \frac{20}{3} - \frac{20}{3}$

$- \frac{26}{3} =- \frac{4}{3}x$

$- \frac{3} {4}\cdot\left (- \frac{26}{3} \right )=- \frac{3} {4}\cdot \left (- \frac{4}{3}x \right )$

$x= \frac{26}{4} = \frac{13}{2} = 6\frac{1}{2}$

The -intercept is the point $\left (6 \frac{1}{2}, 0 \right )$ .

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Question

What is the slope of the line with the equation

Answer

To find the slope, put the equation in the form of .

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

Since $m=-\frac{2}{3}$ , that is the value of the slope.

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Question

A line has the equation . What is the slope of this line?

Answer

You need to put the equation in form before you can easily find out its slope.

$y=\frac{5}{9}x-\frac{4}{3}$

Since $m=\frac{5}{9}$ , that must be the slope.

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Question

The equation of a line is . Find the slope of this line.

Answer

To find the slope, you will need to put the equation in form. The value of will be the slope.

Subtract from either side:

Divide each side by :

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

You can now easily identify the value of .

$m=\frac{3}{2}$

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Question

Find the y-intercept:

Answer

Rewrite the equation in slope-intercept form, .

The y-intercept is , which is .

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Question

Using the similar triangles, find the equation of the line in the provided graph.

Answer

The equation for a line can be written in the slope-intercept form:

,

In this equation, the variables and are defined as the following:

$\textup{m=slope}$

$\textup{b=y-intercept}$

One way to find the slope of a line is to solve for the rise over run:

$\frac{\textup{rise}}{\textup{run}}=\frac{\Delta y}{\Delta x}$

This is defined as the change in the y-axis over the change in the x axis.

$\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

The triangles in the graph provide possess two different values for their respective rise over run calculations; however, both triangles should have the same slope:

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

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

Now that we've found the slope of our line, , we can look at the graph to see where the line crosses the y-axis. The line crosses the y-axis at the following point:

Therefore, the equation of this line is,

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Question

Using the similar triangles, find the equation of the line in the provided graph.

Answer

The equation for a line can be written in the slope-intercept form:

,

In this equation, the variables and are defined as the following:

$\textup{m=slope}$

$\textup{b=y-intercept}$

One way to find the slope of a line is to solve for the rise over run:

$\frac{\textup{rise}}{\textup{run}}=\frac{\Delta y}{\Delta x}$

This is defined as the change in the y-axis over the change in the x axis.

$\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

The triangles in the graph provide possess two different values for their respective rise over run calculations; however, both triangle should have the same slope:

$\frac{3}{3}=1$

$\frac{7}{7}=1$

Now that we've found the slope of our line, , we can look at the graph to see where the line crosses the y-axis. The line crosses the y-axis at the following point:

Therefore, the equation of this line is,

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Question

Using the similar triangles, find the equation of the line in the provided graph.

Answer

The equation for a line can be written in the slope-intercept form:

,

In this equation, the variables and are defined as the following:

$\textup{m=slope}$

$\textup{b=y-intercept}$

One way to find the slope of a line is to solve for the rise over run:

$\frac{\textup{rise}}{\textup{run}}=\frac{\Delta y}{\Delta x}$

This is defined as the change in the y-axis over the change in the x axis.

$\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

The triangles in the graph provide possess two different values for their respective rise over run calculations; however, both triangle should have the same slope:

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

$\frac{3}{6}=\frac{1}{2}$

Now that we've found the slope of our line, $\frac{1}{2}$ , we can look at the graph to see where the line crosses the y-axis. The line crosses the y-axis at the following point:

Therefore, the equation of this line is,

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

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Question

Using the similar triangles, find the equation of the line in the provided graph.

Answer

The equation for a line can be written in the slope-intercept form:

,

In this equation, the variables and are defined as the following:

$\textup{m=slope}$

$\textup{b=y-intercept}$

One way to find the slope of a line is to solve for the rise over run:

$\frac{\textup{rise}}{\textup{run}}=\frac{\Delta y}{\Delta x}$

This is defined as the change in the y-axis over the change in the x axis.

$\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

The triangles in the graph provide possess two different values for their respective rise over run calculations; however, both triangle should have the same slope:

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

$\frac{4}{8}=\frac{1}{2}$

Now that we've found the slope of our line, $\frac{1}{2}$ , we can look at the graph to see where the line crosses the y-axis. The line crosses the y-axis at the following point:

Therefore, the equation of this line is,

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

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Question

Using the similar triangles, find the equation of the line in the provided graph.

Answer

The equation for a line can be written in the slope-intercept form:

,

In this equation, the variables and are defined as the following:

$\textup{m=slope}$

$\textup{b=y-intercept}$

One way to find the slope of a line is to solve for the rise over run:

$\frac{\textup{rise}}{\textup{run}}=\frac{\Delta y}{\Delta x}$

This is defined as the change in the y-axis over the change in the x axis.

$\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

The triangles in the graph provide possess two different values for their respective rise over run calculations; however, both triangle should have the same slope:

$\frac{1}{3}=\frac{1}{3}$

$\frac{2}{6}=\frac{1}{3}$

Now that we've found the slope of our line, $\frac{1}{3}$ , we can look at the graph to see where the line crosses the y-axis. The line crosses the y-axis at the following point:

Therefore, the equation of this line is,

$y=\frac{1}{3}x$

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