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HSPT Math : Concepts

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Example Questions

Example Question #1 : How To Find The Endpoints Of A Line Segment

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 ?

Possible Answers:

B + 8

2B + 20

2B + 1

B + 1

2B + 8

Correct answer:

2B + 8

Explanation:

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:

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

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

Y + 6 = 2B + 14

Y + 6 -6 = 2B + 14 -6

Y = 2B + 8

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Example Question #2 : How To Find The Endpoints Of A Line Segment

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 ?

Possible Answers:

2 A + 2B+8

A + B- 16

2 A + 2B- 8

A + B- 8

A + B+8

Correct answer:

2 A + 2B- 8

Explanation:

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 :

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

T + 8 =2 A + 2B

T + 8 - 8 =2 A + 2B- 8

T =2 A + 2B- 8

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Example Question #1 : How To Find Out If Lines Are Perpendicular

Two perpendicular lines intersect at the point (5,4) . One line passes through point (3,1) ; the other passes through point $\left ( 4, Y\right )$ . Evaluate .

Possible Answers:

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

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

$Y = 5\frac{1}{2}$

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

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

Correct answer:

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

Explanation:

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 (5,4) , 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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Example Question #3 : Use Similar Triangles To Show Equal Slopes: Ccss.Math.Content.8.Ee.B.6

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

Possible Answers:

$\left (0,2 \right )$

$\left (0,0 \right )$

$\left ( 0, -3 \frac{1}{2}\right )$

$\left (0,5 \right )$

$\left (0,-7 \right )$

Correct answer:

$\left (0,5 \right )$

Explanation:

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

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

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

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

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

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Example Question #2 : X And Y Intercept

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

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

Possible Answers:

$\left ( 0, \frac{3}{2}\right )$

(0,4)

The graph has no -intercept.

(0,2)

(0,3)

Correct answer:

The graph has no -intercept.

Explanation:

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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Example Question #2 : How To Find X Or Y Intercept

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

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

Possible Answers:

The graph has no -intercept.

$\left ( 0, \frac{3}{8}\right )$

$\left ( 0, \frac{3}{5}\right )$

$\left ( 0, -\frac{3}{8}\right )$

$\left ( 0,- \frac{3}{5}\right )$

Correct answer:

$\left ( 0, \frac{3}{8}\right )$

Explanation:

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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Example Question #4 : X And Y Intercept

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

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

Possible Answers:

(0, 8)

(0, -64)

(0, -42)

The graph does not have a -intercept.

(0, -16)

Correct answer:

(0, -16)

Explanation:

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:

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

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

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

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

y = 12+ 4 ( -7)

y = 12+ (-28)

y = -16

The -intercept is the point (0, -16) .

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Example Question #1 : Coordinate Geometry

Give the slope of the line that passes through (5,3) and (-1, 5) .

Possible Answers:

$\frac{4}{3}$

$\frac{1}{3}$

$-\frac{1}{3}$

Correct answer:

$-\frac{1}{3}$

Explanation:

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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Example Question #2 : Coordinate Geometry

What is the slope of the line that passes through (4,4) and (8,8) ?

Possible Answers:

0.5

2.5

1.5

Correct answer:

Explanation:

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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Example Question #3 : Coordinate Geometry

What is the slope of the line that passes through (4,7) and (-3,-7) ?

Possible Answers:

1.5

3.5

Correct answer:

Explanation:

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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HSPT Math : Concepts

Study concepts, example questions & explanations for HSPT Math

All HSPT Math Resources

Example Questions

Example Question #1 : How To Find The Endpoints Of A Line Segment

Example Question #2 : How To Find The Endpoints Of A Line Segment

Example Question #1 : How To Find Out If Lines Are Perpendicular

Example Question #3 : Use Similar Triangles To Show Equal Slopes: Ccss.Math.Content.8.Ee.B.6

Example Question #2 : X And Y Intercept

Example Question #2 : How To Find X Or Y Intercept

Example Question #4 : X And Y Intercept

Example Question #1 : Coordinate Geometry

Example Question #2 : Coordinate Geometry

Example Question #3 : Coordinate Geometry

All HSPT Math Resources

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