Consider the function y equals 2 x squared plus 4 x minus 28y=2x2+4x−28.

Question

​(a) Give the coordinates of the vertex of the graph of the function.
​(b) Graph the function on a window that includes the vertex.
Question content area bottom
Part 1
​(a) The vertex of the graph of the function is at the point enter your response here. ​(Type an ordered​ pair.)
Part 2
​(b) Choose the correct graph below. Each graph shown below has a window of ​[minus−​10, ​10] by ​[minus−​40, ​10].
A.


A coordinate system has a horizontal x-axis from negative 10 to 10 in increments of 1 and a vertical y-axis from negative 40 to 10 in increments of 5. A downward U-shaped curve has a maximum on the positive x-axis and exits the viewing window in the third and fourth quadrants.
B.


A coordinate system has a horizontal x-axis from negative 10 to 10 in increments of 1 and a vertical y-axis from negative 40 to 10 in increments of 5. A curve rises from left to right at a decreasing rate, passing through the negative x-axis. A second curve is to the right of the first curve and falls at an increasing rate through the positive x-axis.
C.


A coordinate system has a horizontal x-axis from negative 10 to 10 in increments of 1 and a vertical y-axis from negative 40 to 10 in increments of 5. An upward U-shaped curve has a minimum in the third quadrant and passes through the positive and negative x-axis.
D.

Answer 1

To find the vertex of the function \( y = 2x^2 + 4x - 28 \) and the appropriate graph, let's solve the problems step by step.

Part 1: Finding the Vertex

The vertex of a quadratic function \( y = ax^2 + bx + c \) can be found using the formula: \[ x = -\frac{b}{2a} \] Here, \( a = 2 \), \( b = 4 \), and \( c = -28 \).

Calculating the x-coordinate of the vertex: \[ x = -\frac{4}{2 \cdot 2} = -\frac{4}{4} = -1 \]

Next, we need to find the y-coordinate by substituting \( x = -1 \) back into the function: \[ y = 2(-1)^2 + 4(-1) - 28 \] \[ y = 2(1) - 4 - 28 \] \[ y = 2 - 4 - 28 = -30 \]

So the coordinates of the vertex are: \[ (-1, -30) \]

Part 2: Choosing the Correct Graph

Now, let's analyze the possible graphs. The function \( y = 2x^2 + 4x - 28 \) is a parabola that opens upwards (since \( a = 2 > 0 \)) and has its vertex at \( (-1, -30) \).

We need to identify the graph that represents a parabola that opens upwards and that includes the vertex at \( (-1, -30) \).

• Graph A suggests a downward U-shaped curve, which does not represent our function.
• Graph B appears to depict two separate curves, which doesn't match our quadratic function either.
• Graph C represents an upward U-shaped curve. If its vertex lies in the third quadrant (which it does, since the vertex is at (-1,-30)), and since it passes through both the positive and negative x-axis, it seems to fit our quadratic.
• Graph D is not described, so we can't assess it.

Based on our analysis, the correct answer is:

• Vertex: \((-1, -30)\)
• Graph: C (the upward U-shaped curve with a minimum in the third quadrant).