5.

Question

Determine the value of the constant term of the quadratic function in standard form, given its graph.



(1 point)
The value of c in the standard form of the quadratic function y = ax2+bx+c
is
Question 8
6.

In the xy-coordinate plane, the graph of the equations y = 3x2 −12x −36
has zeros at x = and x =,

where < .

The graph has a minimum at ( , -48).

(3 points)
What are the values of

=
​​​​​​​

=

and

=

Answer 1

To answer your questions, let's break them down step by step.

Question 5

The constant term \( c \) in the standard form of the quadratic function \( y = ax^2 + bx + c \) is the value of \( y \) when \( x = 0 \). This term affects the vertical position of the graph.

Since we don't have the graph to provide specific values, if you have it, simply identify the \( y \)-intercept on the graph to find \( c \).

Question 6

To find the zeros of the quadratic function \( y = 3x^2 - 12x - 36 \), we can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Where:

• \( a = 3 \)
• \( b = -12 \)
• \( c = -36 \)

1. Calculate the discriminant:

\[ b^2 - 4ac = (-12)^2 - 4(3)(-36) = 144 + 432 = 576 \]

2. Now find the zeros:

\[ x = \frac{-(-12) \pm \sqrt{576}}{2(3)} = \frac{12 \pm 24}{6} \]

This gives us two values:

• \( x_1 = \frac{12 + 24}{6} = \frac{36}{6} = 6 \)
• \( x_2 = \frac{12 - 24}{6} = \frac{-12}{6} = -2 \)

So, the zeros are:

• \( x = -2 \)
• \( x = 6 \)

Minimum Point

To find the vertex (minimum) of the quadratic function, we can use the vertex formula:

\[ x = -\frac{b}{2a} \]

For our function:

\[ x = -\frac{-12}{2(3)} = \frac{12}{6} = 2 \]

Now substitute \( x = 2 \) back into the function to find the \( y \)-coordinate:

\[ y = 3(2)^2 - 12(2) - 36 = 3(4) - 24 - 36 = 12 - 24 - 36 = -48 \]

Thus, the vertex is at \( (2, -48) \).

Final Answers

1. The zeros are:

   • \( x = -2 \) and \( x = 6 \).

2. The minimum point is:

   • \( (2, -48) \).

So, in the format requested:

• \( x_1 = -2 \)
• \( x_2 = 6 \)
• Minimum point \( (2, -48) \) implies \( = 2 \) and the minimum value (the second coordinate) is \( = -48 \).

Answer 2

5.

Determine the value of the constant term of the quadratic function in standard form, given its graph.



(1 point)
The value of c in the standard form of the quadratic function y = ax2+bx+c
is

Answer 3

To determine the value of the constant term \( c \) in the standard form of the quadratic function \( y = ax^2 + bx + c \), you need to look at the graph of the quadratic function. The constant term \( c \) corresponds to the \( y \)-intercept of the graph, which is the point where the graph intersects the \( y \)-axis (where \( x = 0 \)).

Steps to find \( c \):

1. Identify the \( y \)-intercept: Look at the graph of the quadratic equation and find the point where it crosses the \( y \)-axis. This point will have coordinates of the form \( (0, c) \).
2. Read the \( y \)-value at this point: The value of \( c \) is simply the \( y \)-coordinate of the intersection point.

If you can provide the specific graph or describe its key points (like the \( y \)-intercept), I can help you find the exact value of \( c \). If you have a graph in mind, please share any relevant information!