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SSAT Upper Level Math : How to find the equation of a curve

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

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

Possible Answers:

y=x-6

y=x-3

-2x-y=6

y=x+6

x-2y=-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:

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

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

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

Insufficient information is given to determine the equation.

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

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 #3 :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:

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

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

Insufficient information is given to determine the equation.

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

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

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 #4 :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:

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

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

Insufficient information is given to determine the equation.

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

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

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 #5 :How To Find The Equation Of A Curve

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

Give its equation.

Possible Answers:

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

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

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

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

Insufficient information is given to determine the equation.

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 ofandcannot be determined for certain. The correct response is that insufficient information is given.

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Example Question #6 :How To Find The Equation Of A Curve

Ellipse 1

Give the equation of the above ellipse.

Possible Answers:

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

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

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

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

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

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 lengthis

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

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 #7 :How To Find The Equation Of A Curve

Ellipse 1

Give the equation of the above ellipse.

Possible Answers:

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

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

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

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

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

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 lengthis

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

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 #8 :How To Find The Equation Of A Curve

Ellipse 1

Give the equation of the above ellipse.

Possible Answers:

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

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

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

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

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

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 lengthis

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

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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Example Question #9 :How To Find The Equation Of A Curve

The-intercept and the only-intercept of a vertical parabola on the coordinate plane coincide with the-intercept and the拦截线的方程 3x+ 5y = 30 . Give the equation of the parabola.

Possible Answers:

$y=\frac{5}{18}x^{2}- \frac{10}{3}x+10$

Insufficient information is given to determine the equation.

$y = \frac{50}{3} x^{2}-\frac{1,000}{3}x+ \frac{5,000}{3}$

$y = \frac{3}{50} x^{2}-\frac{6}{5}x+6$

$y = \frac{18}{5} x^{2}-\frac{216}{5}x+ \frac{648}{5}$

Correct answer:

$y = \frac{3}{50} x^{2}-\frac{6}{5}x+6$

Explanation:

To find the-intercept, that is, the point of intersection with the-axis, of the line of equation 3x+ 5y = 30 , set x=0 and solve for:

3 (0)+ 5y = 30

5y = 30

y = 6

The-intercept is (0,6) .

The-intercept can be found by doing the opposite:

3x+ 5 (0) = 30

3x = 30

x=10

The-intercept is (10,0) .

The parabola has these intercepts as well. Also, since the vertical parabola has only one-intercept, 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 = 10,k=0 :

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

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

for some real. To find it, use the-intercept, setting x=0, y = 6

$6 = a(0-10)^{2}$

$6 = a( -10)^{2}$

6 = 100a

$a = \frac{6}{100} = \frac{3}{50}$

The parabola has equation $y =\frac{3}{50} (x-10)^{2}$ , which is rewritten as

$y =\frac{3}{50} (x^{2}-20x+100)$

$y = \frac{3}{50} x^{2}-\frac{6}{5}x+6$

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Example Question #10 :How To Find The Equation Of A Curve

An ellipse on the coordinate plane has as its center the point (-4, 9) . It passes through the points (-4, 5) and (11, 9) . Give its equation.

Possible Answers:

Insufficient information is given to determine the equation.

$\frac{(x+4)^{2}}{225} + \frac{(y-9)^{2}}{16} = 1$

$\frac{(x-4)^{2}}{900} + \frac{(y+9)^{2}}{64} = 1$

$\frac{(x-4)^{2}}{225} + \frac{(y+9)^{2}}{16} = 1$

$\frac{(x+4)^{2}}{900} + \frac{(y-9)^{2}}{64} = 1$

Correct answer:

$\frac{(x+4)^{2}}{225} + \frac{(y-9)^{2}}{16} = 1$

Explanation:

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

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

The center is (-4, 9) , so h = -4 and k = 9 .

To find, note that one endpoint of the horizontal axis is given by the point with the same-coordinate through which it passes, namely, (11, 9) . Half the length of this axis, which is, is the difference of the-coordinates, so a = 11- (-4) = 15 . Similarly, to find, note that one endpoint of the vertical axis is given by the point with the same-coordinate through which it passes, namely, (-4, 5) . Half the length of this axis, which is, is the difference of the-coordinates, so b = 9 - 5 = 4 .

The equation is

$\frac{(x-(-4))^{2}}{15^{2}} + \frac{(y-9)^{2}}{4^{2}} = 1$

$\frac{(x+4)^{2}}{225} + \frac{(y-9)^{2}}{16} = 1$ .

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SSAT Upper Level Math : How to find the equation of a curve

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

All SSAT Upper Level Math Resources

Example Questions

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

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

Example Question #3 :How To Find The Equation Of A Curve

Example Question #4 :How To Find The Equation Of A Curve

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

Example Question #6 :How To Find The Equation Of A Curve

Example Question #7 :How To Find The Equation Of A Curve

Example Question #8 :How To Find The Equation Of A Curve

Example Question #9 :How To Find The Equation Of A Curve

Example Question #10 :How To Find The Equation Of A Curve

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