Use the method of completing the square to write the equation in standard form. Then graph the equation.
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Question

Quizplus · Accepted Answer

The given equation is y = -4x^2 + 12x + 3. To complete the square, we need to add and subtract the square of half of the coefficient of x i.e. (12/(-8))^2 = 9/4.

y = -4(x^2 - (12/(-8))x) + 3
   = -4(x^2 - (3/2)x + 9/16 - 9/16) + 3
   = -4((x - 3/4)^2 - 9/16) + 3
   = -4(x - 3/4)^2 + 57/16

Now the equation is in standard form, y = ax^2 + bx + c, where a = -4, b = 0, and c = 57/16.

To graph this equation, we can find the vertex (h, k) using h = -b/2a and k = c - ah^2.

h = -(0)/(2(-4)) = 0
k = 57/16 - (-4)(0)^2 = 57/16

The vertex is (0, 57/16), which is the highest point on the parabola since a is negative. We can also find the x-intercepts by setting y = 0 and solving for x.

0 = -4(x - 3/4)^2 + 57/16
(x - 3/4)^2 = 57/(4*4)
x - 3/4 = +/-sqrt(57)/(4*2)
x = 3/4 +/- sqrt(57)/8

The x-intercepts are approximately -0.53 and 1.78.

Therefore, the graph will be a downward-facing parabola with vertex at (0, 57/16) and x-intercepts at approximately -0.53 and 1.78.

Choice C is the correct answer. We can complete the square as follows:

y = 2x^2 + 8x + 7

y = 2(x^2 + 4x) + 7

y = 2(x^2 + 4x + 4 - 4) + 7

y = 2((x+2)^2 - 4) + 7

y = 2(x+2)^2 - 1

So the equation in standard form is y = 2(x+2)^2 - 1.

The graph is a parabola that opens upward, with its vertex at (-2,-1), and a vertical stretch factor of 2.

Use the Method of Completing the Square to Write the Equation