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SSAT Upper Level Math : Geometry

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Example Questions

Example Question #221 : Coordinate Geometry

Define a function $f(x) = x ^{2}+ 3x + 2$ . Which of the following is an -intercept of the graph of ?

(a) (2, 0)

(b) (3, 0)

Possible Answers:

Both (a) and (b)

(a), but not (b)

(b), but not (a)

Neither (a) nor (b)

Correct answer:

Neither (a) nor (b)

Explanation:

An -intercept of the graph of a function has 0 as its -coordinate, since it is defined to be a point at which it crosses the -axis. Its -coordinate is a value of for which f(x) = 0 .

We can most easily determine whether (2, 0) is a point on the graph of f(x) by proving or disproving that f(2) = 0 , which we can do by substituting 2 for :

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

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

f(2) = 4+ 6 + 2

f(2) = 1 2

$f(2) \ne 0$ , so (2, 0) is not an -intercept.

Similarly, substituting 3 for :

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

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

f(3) = 9+ 9 + 2

f(3) = 1 2

$f(3) \ne 0$ , so (3, 0) is not an -intercept.

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Example Question #281 : Geometry

Define f(x) = 2x+ 7 . The graphs of f(x) and a second function, g(x) , intersect at their common -intercept. Which of the following could be the definition of g(x) ?

Possible Answers:

g(x) = 2x+ 14

g(x) = 4x+ 14

g(x) = 4x+ 7

g(x) = 7

Correct answer:

g(x) = 4x+ 14

Explanation:

An -intercept of the graph of a function has 0 as its -coordinate, since it is defined to be a point at which it crosses the -axis. Its -coordinate is a value of for which f(x) = 0 , which can be found as follows:

Substituting the definition, we get

2x+ 7 = 0

Solving for by subtracting 7 from both sides, then dividing both sides by 2:

2x+ 7 -7 = 0 -7

2x = -7

$2x \div 2 = -7 \div 2$

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

The -intercept of the graph of f(x) is the point $\left ( - \frac{7}{2}, 0 \right )$ .

To determine which of the four choices is correct, substitute $- \frac{7}{2}$ for and determine for which definition of it holds that $g \left ( - \frac{7}{2} \right ) = 0$ .

g(x) = 7 can be eliminated immediately as a choice since it cannot take the value 0.

g(x) = 2x+ 14

$g\left ( - \frac{7}{2} \right ) = 2\left ( - \frac{7}{2} \right ) + 14 = -7 + 14 = 7$

g(x) = 4x+ 7 :

$g \left ( - \frac{7}{2} \right ) = 4 \left ( - \frac{7}{2} \right ) + 7 = -14 + 7 = -7$

g(x) = 4x+ 14

$g \left ( - \frac{7}{2} \right ) = 4 \left ( - \frac{7}{2} \right ) + 14 = -14 + 14 =0$

The correct choice is g(x) = 4x+ 14 .

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Example Question #1 : How To Find The Equation Of A Curve

If the -intercept of the line is and the slope is , which of the following equations best satisfies this condition?

Possible Answers:

x-2y=-6

-2x-y=6

y=x+6

y=x-3

y=x-6

Correct answer:

y=x-6

Explanation:

Write the slope-intercept form.

y=mx+b

The point given the x-intercept of 6 is (6,0) .

Substitute the point and the slope into the equation and solve for the y-intercept.

0=(1)(6)+b

b=-6

Substitute the y-intercept back to the slope-intercept form to get your equation.

y=x-6

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Example Question #2 : How To Find The Equation Of A Curve

A vertical parabola on the coordinate plane has vertex (-4, 10) and -intercept (0, -2) .

Give its equation.

Possible Answers:

$y = \frac{3}{4} x^{2} + 6x+22$

$y = -\frac{3}{4} x^{2}- 6x-2$

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

Insufficient information is given to determine the equation.

$y = \frac{1}{2} x^{2}+4x+18$

Correct answer:

$y = -\frac{3}{4} x^{2}- 6x-2$

Explanation:

The equation of a vertical parabola, in vertex form, is

$y = a(x-h)^{2}+k$ ,

where (h,k) is the vertex. Set h = -4,k=10 :

$y = a[x-(-4)]^{2}+10$

$y = a(x+4)^{2}+10$

To find , use the -intercept, setting x=0,y= -2 :

$-2 = a(0+4)^{2}+10$

$-2 = a\cdot 4^{2}+10$

-2 = 16a +10

-12 = 16a

$a =- \frac{3}{4}$

The equation, in vertex form, is $y =- \frac{3}{4}(x+4)^{2}+10$ ; in standard form:

$y = -\frac{3}{4}(x+4)^{2}+10$

$y =- \frac{3}{4}(x^{2} + 8x+16)+10$

$y = -\frac{3}{4} x^{2}- 6x-12+10$

$y = -\frac{3}{4} x^{2}- 6x-2$

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Example Question #1 : How To Find The Equation Of A Curve

A vertical parabola on the coordinate plane has vertex (-3, -6) ; one of its -intercepts is (-7, 0) .

Give its equation.

Possible Answers:

$y = 3x^{2}+36x+105$

$y =- \frac{3}{8}x^{2}- \frac{9}{4}x - \frac{75}{8}$

$y =- 3x^{2}-36x-111$

Insufficient information is given to determine the equation.

$y = \frac{3}{8}x^{2}+ \frac{9}{4}x - \frac{21}{8}$

Correct answer:

$y = \frac{3}{8}x^{2}+ \frac{9}{4}x - \frac{21}{8}$

Explanation:

The equation of a vertical parabola, in vertex form, is

$y = a(x-h)^{2}+k$ ,

where (h,k) is the vertex. Set h = -3,k=-6 :

$y = a[x-(-3)]^{2}+ (-6)$

$y = a(x+3)^{2}-6$

To find , use the known -intercept, setting x= -7, y=0 :

$0 = a(-7+3)^{2}-6$

$0 = a(-4)^{2}-6$

0 =16 a-6

16a= 6

$a = \frac{6}{16} = \frac{3}{8}$

The equation, in vertex form, is $y = \frac{3}{8}(x+3)^{2}-6$ ; in standard form:

$y = \frac{3}{8}(x+3)^{2}-6$

$y = \frac{3}{8}(x^{2}+6x+9)-6$

$y = \frac{3}{8}x^{2}+ \frac{9}{4}x+ \frac{27}{8}-6$

$y = \frac{3}{8}x^{2}+ \frac{9}{4}x+ \frac{27}{8}- \frac{48}{8}$

$y = \frac{3}{8}x^{2}+ \frac{9}{4}x - \frac{21}{8}$

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Example Question #2 : How To Find The Equation Of A Curve

A vertical parabola on the coordinate plane has -intercept (0,4) ; its only -intercept is (6,0) .

Give its equation.

Possible Answers:

$y =- \frac{1}{6} x^{2}+\frac{1}{3}x+4$

Insufficient information is given to determine the equation.

$y =- \frac{1}{9} x^{2}+4$

$y = \frac{1}{6} x^{2}-\frac{5}{3}x+4$

$y = \frac{1}{9} x^{2}-\frac{4}{3}x+4$

Correct answer:

$y = \frac{1}{9} x^{2}-\frac{4}{3}x+4$

Explanation:

If a vertical parabola has only one -intercept, which here is (6,0) , that point doubles as its vertex as well.

The equation of a vertical parabola, in vertex form, is

$y = a(x-h)^{2}+k$ ,

where (h,k) is the vertex. Set h = 6,k=0 :

$y = a(x-6)^{2}+0$

$y = a(x-6)^{2}$

To find , use the -intercept, setting x= 0, y=4 :

$4= a(0-6)^{2}$

$4= a( -6)^{2}$

4 = 36a

$a = \frac{4}{36} = \frac{1}{9}$

The equation, in vertex form, is $y = \frac{1}{9}(x-6)^{2}$ . In standard form:

$y = \frac{1}{9}(x-6)^{2}$

$y = \frac{1}{9}(x^{2}-12+36)$

$y = \frac{1}{9} x^{2}-\frac{4}{3}x+4$

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Example Question #282 : Geometry

A vertical parabola on the coordinate plane has -intercept (0, 6) ; one of its -intercepts is (2,0) .

Give its equation.

Possible Answers:

Insufficient information is given to determine the equation.

$y =2 x^{2}-7 x + 6$

$y = x^{2}-5 x + 6$

$y = - x^{2}- x + 6$

$y = -2 x^{2}+ x + 6$

Correct answer:

Insufficient information is given to determine the equation.

Explanation:

The equation of a vertical parabola, in standard form, is

$y = ax^{2}+ bx + c$

for some real a,b,c .

is the -coordinate of the -intercept, so c=6 , and the equation is

$y = ax^{2}+ bx + 6$

Set x=2, y=0 :

$0 = a\cdot 2^{2}+ b \cdot 2 + 6$

0 = 4a+2b + 6

However, no other information is given, so the values of and cannot be determined for certain. The correct response is that insufficient information is given.

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Example Question #21 : X And Y Intercept

Ellipse 1

Give the equation of the above ellipse.

Possible Answers:

$\frac{(x+6)^{2}}{16} + \frac{(y+2)^{2}}{64} = 1$

$\frac{(x-6)^{2}}{4} + \frac{(y-2)^{2}}{8} = 1$

$\frac{(x+6)^{2}}{4} + \frac{(y+2)^{2}}{8} = 1$

$\frac{(x-6)^{2}}{16} + \frac{(y-2)^{2}}{64} = 1$

$\frac{(x+6)^{2}}{64} + \frac{(y+2)^{2}}{256} = 1$

Correct answer:

$\frac{(x-6)^{2}}{16} + \frac{(y-2)^{2}}{64} = 1$

Explanation:

The equation of the ellipse with center (h,k) , horizontal axis of length , and vertical axis of length is

$\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1$

The ellipse has center (6,2) , horizontal axis of length 8, and vertical axis of length 16. Therefore,

h = 6,k = 2 , $a = \frac{8}{2} = 4$ , and $b = \frac{16}{2} = 8$ .

The equation of the ellipse is

$\frac{(x-6)^{2}}{4^{2}} + \frac{(y-2)^{2}}{8^{2}} = 1$

$\frac{(x-6)^{2}}{16} + \frac{(y-2)^{2}}{64} = 1$

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Example Question #5 : How To Find The Equation Of A Curve

Ellipse 1

Give the equation of the above ellipse.

Possible Answers:

$\frac{(x-1)^{2}}{5} + \frac{(y-1)^{2}}{3} = 1$

$\frac{(x-1)^{2}}{25} + \frac{(y-1)^{2}}{9} = 1$

$\frac{(x-1)^{2}}{100} + \frac{(y-1)^{2}}{36} = 1$

$\frac{(x+1)^{2}}{25} + \frac{(y+1)^{2}}{9} = 1$

$\frac{(x+1)^{2}}{100} + \frac{(y+1)^{2}}{36} = 1$

Correct answer:

$\frac{(x-1)^{2}}{25} + \frac{(y-1)^{2}}{9} = 1$

Explanation:

The equation of the ellipse with center (h,k) , horizontal axis of length , and vertical axis of length is

$\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1$

The ellipse has center (1,1) , horizontal axis of length 10, and vertical axis of length 6. Therefore,

h = k = 1 , $a = \frac{10}{2} = 5$ , and $b = \frac{6}{2} = 3$ .

The equation of the ellipse is

$\frac{(x-1)^{2}}{5^{2}} + \frac{(y-1)^{2}}{3^{2}} = 1$

$\frac{(x-1)^{2}}{25} + \frac{(y-1)^{2}}{9} = 1$

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Example Question #22 : X And Y Intercept

Ellipse 1

Give the equation of the above ellipse.

Possible Answers:

$\frac{(x+3)^{2}}{4} + \frac{(y+5)^{2}}{3} = 1$

$\frac{(x-3)^{2}}{16} + \frac{(y-5)^{2}}{9} = 1$

$\frac{(x+3)^{2}}{16} + \frac{(y+5)^{2}}{9} = 1$

$\frac{(x-3)^{2}}{4} + \frac{(y-5)^{2}}{3} = 1$

$\frac{(x-3)^{2}}{8} + \frac{(y-5)^{2}}{6} = 1$

Correct answer:

$\frac{(x-3)^{2}}{16} + \frac{(y-5)^{2}}{9} = 1$

Explanation:

The equation of the ellipse with center (h,k) , horizontal axis of length , and vertical axis of length is

$\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1$

The ellipse has center (3,5) , horizontal axis of length 8, and vertical axis of length 6. Therefore,

h = 3,k = 5 , $a = \frac{8}{2} = 4$ , and $b = \frac{6}{2} = 3$ .

The equation of the ellipse is

$\frac{(x-3)^{2}}{4^{2}} + \frac{(y-5)^{2}}{3^{2}} = 1$

$\frac{(x-3)^{2}}{16} + \frac{(y-5)^{2}}{9} = 1$

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SSAT Upper Level Math : Geometry

Study concepts, example questions & explanations for SSAT Upper Level Math

All SSAT Upper Level Math Resources

Example Questions

Example Question #221 : Coordinate Geometry

Example Question #281 : Geometry

Example Question #1 : How To Find The Equation Of A Curve

Example Question #2 : How To Find The Equation Of A Curve

Example Question #1 : How To Find The Equation Of A Curve

Example Question #2 : How To Find The Equation Of A Curve

Example Question #282 : Geometry

Example Question #21 : X And Y Intercept

Example Question #5 : How To Find The Equation Of A Curve

Example Question #22 : X And Y Intercept

All SSAT Upper Level Math Resources

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