46. Use factoring to rewrite y = 3x² + 8x + 4, so that you can sketch the parabola.
Rewrite the equation in factored form.
b. Locate x-intercepts
C. Determine equation for axis of symmetry
D. Determine vertex
E. Sketch the graph
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To rewrite the equation y = 3x² + 8x + 4 in factored form, follow these steps:
Step 1: Factor out the common factor, if possible.
In this case, the equation does not have a common factor other than 1, so we move on to the next step.
Step 2: Find two numbers whose product is equal to the product of the coefficient of x² (3) and the constant (4), and whose sum is equal to the coefficient of x (8).
The product of 3 and 4 is 12, and the sum of two numbers that multiply to 12 is 6 and 2.
Step 3: Rewrite the equation using the two numbers found in Step 2.
y = 3x² + 6x + 2x + 4
Step 4: Group terms and factor by grouping.
y = (3x² + 6x) + (2x + 4)
y = 3x(x + 2) + 2(x + 2)
Step 5: Factor out the common factor (x + 2).
y = (x + 2)(3x + 2)
Now that we have the factored form of the equation as (x + 2)(3x + 2), we can proceed to answer the rest of the questions:
b. To locate the x-intercepts, we set y = 0 and solve for x:
(x + 2)(3x + 2) = 0
Setting each factor to 0:
x + 2 = 0 -> x = -2
3x + 2 = 0 -> 3x = -2 -> x = -2/3
Therefore, the x-intercepts are x = -2 and x = -2/3.
c. To determine the equation for the axis of symmetry, we take the average of the x-intercepts:
Axis of symmetry: (x + (-2/3) + (-2))/2 = (x - 8/3)/2
d. To determine the vertex, we substitute the x-coordinate of the axis of symmetry into the factored form of the equation and solve for y:
y = (x + 2)(3x + 2)
Substituting x = -8/3 into the equation:
y = (-8/3 + 2)(3(-8/3) + 2)
Simplifying:
y = (-2/3)(-2/3)
y = 4/9
Therefore, the vertex is (-8/3, 4/9).
e. To sketch the graph, plot the vertex, the x-intercepts, and a few additional points on the parabola to get an accurate curve.