How to do coordinate geometry - HSPT Math

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Question

Which of the following is an equation of a line with slope 0?

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Answer

A line with slope 0 has equation for some real value .

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Question

Give the equation of the red line in slope-intercept form.

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Answer

The slope of the line is

$m = \frac{y_{2}- y_{1}}{x_{2}- x_{1}} = \frac{6- (-1)}{0-(-4)} = \frac{7}{4}$

The -intercept of the line has -coordinate

The slope-intercept form can be written:

Replace:

$y = \frac{7}{4}x+6$

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Question

What is the slope of a line through the points and ?

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Answer

Use the slope fomula, setting $x_{1} = -2, x_{2} = 2,y_{1} = 3, y_{2} = 5$

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{5-3}{2-(-2)} = \frac{2}{4} =\frac{1}{2}$

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Question

In which quadrant, or on which axis, is the point with coordinates located?

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Answer

Any point with a positive -coordinate and a negative -coordinate is located in the lower right quadrant - Quadrant IV.

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Question

The green and blue lines are perpendicular: What is the slope of the blue line?

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Answer

The slope of the blue line, being perpendicular to the green line, is the opposite of the reciprocal of the slope of the green line. The slope of the green line can be found using the slope formula:

$m = \frac{y_{2}- y_{1}}{x_{2}- x_{1}} = \frac{10-0}{0-30} = -\frac{1}{3}$

The opposite of the reciprocal of $-\frac{1}{3}$ is 3, and this is the slope of the blue line.

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Question

A line segment on the coordinate plane has endpoints and . In terms of and , as applicable, give the -coordinate of its midpoint.

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Answer

The -coordinate of the midpoint of a line segment is the mean of the -coordinates of its endpoints. Therefore, the -coordinate is

$y= \frac{(a-4)+(-b)}{2} = \frac{a-b-4}{2}$ .

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Question

A line segment on the coordinate plane has endpoints and . In terms of and , as applicable, give the -coordinate of its midpoint.

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Answer

The -coordinate of the midpoint of a line segment is the mean of the -coordinates of its endpoints. Therefore, the -coordinate is

$x= \frac{(a-5)+(b+8)}{2} = \frac{a+b-5+8}{2} = \frac{a+b+3}{2}$ .

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Question

What is the -intercept of the graph of the function

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

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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 slope of the line that passes through and ?

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Answer

We can use the slope formula:

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{8-4}{8-4}=\frac{4}{4}=1$

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Question

A line segment on the coordinate plane has one endpoint at $\left ( 8, 10 \right )$ ; its midpoint is $\left ( A+B, A- B\right )$ . Which of the following gives the -coordinate of its other endpoint in terms of and ?

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Answer

To find the value of the -coordinate of the other endpoint, we will assign the variable . Then, since the -coordinate of the midpoint of the segment is the mean of those of its endpoints, the equation that we can set up is

$\frac{T + 8 }{2} = A + B$ .

We solve for :

$\frac{T + 8 }{2} \cdot 2 =\left ( A + B \right )\cdot 2$

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Question

Two perpendicular lines intersect at the point . One line passes through point ; the other passes through point $\left ( 4, Y\right )$ . Evaluate .

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Answer

The line that passes through $\left ( 3,1\right )$ and $\left ( 5,4\right )$ has slope

$m = \frac{4-1}{5-3} = \frac{3}{2}$ .

The line that passes through $\left ( 4, Y\right )$ and , being perpendicular to the first, has as its slope the opposite reciprocal of $\frac{3}{2}$ , or $-\frac{2}{3}$ .

Therefore, to find , we use the slope formula and solve for :

$-\frac{2}{3} = \frac{Y-4}{4-5}$

$-\frac{2}{3} = \frac{Y-4}{-1}$

$-\frac{2}{3} \cdot (-1) = \frac{Y-4}{-1} \cdot (-1)$

$\frac{2}{3}= Y-4$

$\frac{2}{3}+ 4= Y-4 + 4$

$4\frac{2}{3}= Y$

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Question

A line segment on the coordinate plane has midpoint $\left ( A + 5, B + 7\right )$ . One of its endpoints is $\left ( 10, 6 \right )$ . What is the -coordinate of the other endpoint, in terms of and/or ?

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Answer

Let be the -coordinate of the other endpoint. Since the -coordinate of the midpoint of the segment is the mean of those of the endpoints, we can set up an equation as follows:

$\frac{Y + 6 }{2} = B + 7$

$\frac{Y + 6 }{2} \cdot 2= \left ( B + 7 \right )\cdot 2$

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Question

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

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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 slope of the line that passes through and .

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Answer

Use the slope formula, substituting $x_{1}= 5, y_{1}=3,x_{2}= -1, y_{2}=5$ :

$m =\frac{ y_{2}- y_{1} }{x_{2} - x_{1}}=\frac{ 5-3 }{-1-5} = \frac{ 2 }{-6} =-\frac{1 }{3}$

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Question

What is the slope of the line that passes through and ?

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Answer

We can use the slope formula:

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{-7-7}{-3-4}=\frac{-14}{-7}=2$

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Question

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

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

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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

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

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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

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

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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

A point is shifted $\textup{3 units}$ to the left and $\textup{5 units}$ down. What is the new point?

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Answer

A point is defined as . If the point is shifted left or right, the value will be added or subtracted. If the point is shifted up or down, then the value will be added or subtracted.

3 units to the left is:

5 units down is:

The new point is:

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Question

What is the slope given the points and ?

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Answer

Write the slope equation, substitute the point values, and solve for the slope.

$m=\frac{y_2-y_1}{x_2-x_1}= \frac{5-6}{-4-3}= \frac{-1}{-7}= \frac{1}{7}$

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