What is the equation of the parabola shown below?
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Solution
Create an equation that describes the situation. If x represents the first even integer, x + 2 represents the second integer and x + 4 represents the third. The sum of the integers is x + x + 2 + x + 4 = 3x + 6. Set this expression equal to the sum of the integers, –18, and solve for x: 3x + 6 = –18, 3x = –24, x = –8. Since x is the first number, it is the smallest of the three consecutive, even integers.
Which of the following is the equation of a parabola whose vertex is (5,0)?
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Solution
Create an equation that describes the situation.If x represents the first odd whole number, x + 2 represents the second, x + 4 represents the third, and x+ 6 represents the fourth. The sum of the numbers is x + x + 2 + x + 4 + x + 6 = 4x+ 12.Set this expression equal to the sum of the numbers, 48, and solve for x: 4x + 12 = 48, 4x = 36, x = 9. Since x is the first number, it is the smallest of the four consecutive, odd whole numbers.
What is the vertex of the parabola whose equation is y = (x + 2)2 + 2?
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Solution
Create an equation that describes the situation. If x represents the first integer, x + 1 represents the second integer, and x + 2 represents the third integer. The sum of the integers is x + x + 1 + x + 2 = 3x + 3. Set this expression equal to the sum of the integers, 63, and solve for x: 3x + 3 = 63, 3x = 60, x = 20. Since x is the first integer, 21 is the second integer, and 22 is the third (and largest) integer.
Which of the following is the equation of a parabola whose vertex is (–3,–4)?
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Solution
Create an equation that describes the situation. If x represents the number, then one-fourth the number is 1⁄4(x). Three more than that is 1⁄4(x) + 3.Three less than the number is x – 3. Set these expressions equal to each other, and solve for x: 1⁄4x(x) + 3 = x – 3, 3 = 3⁄4(x) – 3, 6 = 3⁄4(x), x = 8.
What is the vertex of the parabola whose equation is y = x2 + 4?
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Solution
Create an equation that describes the situation. If x represents the number, then 3 times the number is 3x. Five less than that is 3x – 5. Set that expression equal to 10, and solve for x: 3x – 5 = 10, 3x = 15, x = 5.
What values make the fraction \(\frac{(x^{2}-36)}{(2x^{2}-25x+72)}\) undefined?
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Solution
A fraction is undefined when its denominator is equal to 0. Set each term of the denominator equal to 0 and solve for x; 2x(x – 5) = 0; 2x = 0, x = 0 and x – 5 = 0, x = 5. The fraction is undefined when x = 0 or 5.
What is one value that makes the fraction \(\frac{\left ( x^{2} \right )}{\left ( 9x^{2}-1 \right )}\) undefined?
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Solution
A fraction is undefined when its denominator is equal to 0. Set 8 – 8y + 4 equal to 0 and solve for y; 8 – 8y + 4 = 0, –8y + 12 = 0, –8y = –12, y =3⁄2. The fraction is undefined when y =3⁄2.
What is one value that makes the fraction \(\frac{\left ( x-16 \right )}{\left ( x^{2}-16 \right )}\) undefined?
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Solution
A fraction is undefined when its denominator is equal to 0. Set 6a + 18 – 4a equal to 0 and solve for a; 6a + 18 – 4a = 0, 2a + 18 = 0, 2a = –18, a = –9. The fraction is undefined when a = –9.
What values make the fraction \(\frac{\left (x^{2}-8x+7 \right )}{\left (x-8x+7 \right )}\) undefined?
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Solution
A fraction is undefined when its denominator is equal to 0. Set 6d equal to 0 and solve for d; 6d = 0, d = 0. The fraction is undefined when d = 0.
For what value of x is the fraction \(\frac{-8}{x-8}\) undefined?
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Solution
A fraction is undefined when its denominator is equal to 0. Set x – 8 equal to 0 and solve for x; x – 8 = 0, x = 8. The fraction is undefined when x = 8.