y=a(x−h)^2+k and sketch its graph.
y=2x^2+12x+14
y = 2x^2+12x+14
y = 2(x^2+6x+9)-4
y = 2(x+3)^2-4
so, the vertex is at (-3,-4)
the y-intercept is at (0,14)
the x-intercepts are at (-3±√2,0)
To sketch the graph of the quadratic function y = 2x^2 + 12x + 14, you can follow these steps:
Step 1: Find the vertex of the parabola
In the given equation, the quadratic function is in the vertex form y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. The values of h and k can be obtained by converting the equation from standard form to vertex form using the process called completing the square.
y = 2x^2 + 12x + 14
First, factor out the coefficient of x^2 from the x terms:
y = 2(x^2 + 6x) + 14
Next, complete the square utilizing the coefficient of x:
y = 2(x^2 + 6x + 9) - 2(9) + 14
Simplify:
y = 2(x + 3)^2 - 2(9) + 14
y = 2(x + 3)^2 - 18 + 14
y = 2(x + 3)^2 - 4
From this equation, we can see that the vertex is at (-3, -4).
Step 2: Determine the y-intercept
To find the y-intercept, substitute x = 0 into the equation:
y = 2(0)^2 + 12(0) + 14
y = 0 + 0 + 14
y = 14
So, the y-intercept is at (0, 14).
Step 3: Find the x-intercepts (if any)
To find the x-intercepts, set y = 0 and solve for x. Since the equation is in the standard form, we can use factoring, the quadratic formula, or completing the square to solve for x. In this case, we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 2, b = 12, and c = 14.
x = (-12 ± √(12^2 - 4(2)(14))) / (2(2))
x = (-12 ± √(144 - 112)) / 4
x = (-12 ± √32) / 4
x = (-12 ± 4√2) / 4
x = -3 ± √2
So, the x-intercepts are approximately -3 + √2 and -3 - √2.
Step 4: Plot the points and sketch the graph
Now that we have the vertex, y-intercept, and x-intercepts (if any), we can plot these points and sketch the graph. The vertex is at (-3, -4), the y-intercept is at (0, 14), and there are two x-intercepts: -3 + √2 and -3 - √2.
The graph of the quadratic function y = 2x^2 + 12x + 14 will be a U-shaped curve opening upward due to the positive coefficient of x^2. It will pass through the points (-3, -4), (0, 14), and intersect the x-axis at approximately -3 + √2 and -3 - √2.
Note: If you have access to a graphing calculator or online graphing tool, you can also plot the function to get a more accurate graph.