In the xy-coordinate plane, the graph of the equation y = 3x² - 12x - 36 has zeros at x = a and x = b, where a < b. The graph has a minimum at (c,-48). What are the values of a, b, and c?
A.) a = 2, b = 4, c = 2
B.) a = -2, b = 6, c = 2
C.) a = -3, b = 3, c = 0
D.) a = 3, b = 6, c = 2
A.) a = 2, b = 4, c = 2
B.) a = -2, b = 6, c = 2
C.) a = -3, b = 3, c = 0
D.) a = 3, b = 6, c = 2
First, find the zeros of the equation. When finding zeros of a quadratic equation, you are finding the x-intercepts. At the x-intercept, the value of y is 0. To find the x-intercepts, substitute 0 for y.
y = 3x² - 12x - 36
3x² - 12x - 36 = 0 {substituted 0 for y}
3(x² - 4x - 12) = 0 {factored out the greatest common factor, 3}
3(x - 6)(x + 2) = 0 {factored into two binomials}
x - 6 = 0 or x + 2 = 0 {set each factor equal to 0}
x = 6 or x = -2 {solved each equation for x}
It is given that the zeros are at x = a and x = b, and since it is also given that a < b, then
a = -2 and b = 6.
It is given the graph has a minimum at (c,-48). A minimum of a quadratic equation is, graphically, the vertex of the parabola. Since the minimum is at (c,-48), then the y-coordinate of the vertex is -48. To find out what x is when y is -48, substitute -48 in for y into the equation and you will be finding the value of c.
y = 3x² - 12x - 36
-48 = 3x² - 12x - 36 {substituted -48 in for y}
3x² - 12x + 12 = 0 {added 48 to each side}
3(x² - 4x + 4) = 0 {factored out the greatest common factor, 3}
3(x - 2)(x - 2) = 0 {factored into two binomials}
x - 2 = 0 {set each factor equal to 0}
x = 2 {added 2 to each side}
The value of x, when y is -48, is 2. Therefore, the x-coordinate of the vertex is 2, meaning c = 2.
B.) a = -2, b = 6, c = 2
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y = 3x² - 12x - 36
3x² - 12x - 36 = 0 {substituted 0 for y}
3(x² - 4x - 12) = 0 {factored out the greatest common factor, 3}
3(x - 6)(x + 2) = 0 {factored into two binomials}
x - 6 = 0 or x + 2 = 0 {set each factor equal to 0}
x = 6 or x = -2 {solved each equation for x}
It is given that the zeros are at x = a and x = b, and since it is also given that a < b, then
a = -2 and b = 6.
It is given the graph has a minimum at (c,-48). A minimum of a quadratic equation is, graphically, the vertex of the parabola. Since the minimum is at (c,-48), then the y-coordinate of the vertex is -48. To find out what x is when y is -48, substitute -48 in for y into the equation and you will be finding the value of c.
y = 3x² - 12x - 36
-48 = 3x² - 12x - 36 {substituted -48 in for y}
3x² - 12x + 12 = 0 {added 48 to each side}
3(x² - 4x + 4) = 0 {factored out the greatest common factor, 3}
3(x - 2)(x - 2) = 0 {factored into two binomials}
x - 2 = 0 {set each factor equal to 0}
x = 2 {added 2 to each side}
The value of x, when y is -48, is 2. Therefore, the x-coordinate of the vertex is 2, meaning c = 2.
B.) a = -2, b = 6, c = 2
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