Question

)What are the key points on the graph of y=x2−16x+28
? Name the vertex, x-intercepts, and y-intercept.(2 points)
vertex: (
)

(smaller) x-intercept: (
)

(larger) x-intercept: (
)

y-intercept: (
)

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Answer 1

To analyze the quadratic function \( y = x^2 - 16x + 28 \), we will find the vertex, x-intercepts (roots), and y-intercept.

### Step 1: Finding the Vertex

The vertex of a quadratic function in the form \( y = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex:

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

In this case, \( a = 1 \) and \( b = -16 \). Plugging in these values:

\[
x = -\frac{-16}{2 \cdot 1} = \frac{16}{2} = 8
\]

Now, we substitute \( x = 8 \) into the original equation to find the y-coordinate:

\[
y = (8)^2 - 16(8) + 28 = 64 - 128 + 28 = -36
\]

So, the vertex is:

\[
\text{vertex: } (8, -36)
\]

### Step 2: Finding the x-intercepts

To find the x-intercepts, we set \( y = 0 \):

\[
0 = x^2 - 16x + 28
\]

This is a quadratic equation, and we can solve it using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Here, \( a = 1 \), \( b = -16 \), and \( c = 28 \). We calculate the discriminant:

\[
b^2 - 4ac = (-16)^2 - 4 \cdot 1 \cdot 28 = 256 - 112 = 144
\]

Now, apply the quadratic formula:

\[
x = \frac{-(-16) \pm \sqrt{144}}{2 \cdot 1} = \frac{16 \pm 12}{2}
\]

Calculating the two values:

1. \( x = \frac{16 + 12}{2} = \frac{28}{2} = 14 \)
2. \( x = \frac{16 - 12}{2} = \frac{4}{2} = 2 \)

Thus, the x-intercepts are:

\[
\text{smaller x-intercept: } (2, 0)
\]
\[
\text{larger x-intercept: } (14, 0)
\]

### Step 3: Finding the y-intercept

The y-intercept occurs when \( x = 0 \):

\[
y = (0)^2 - 16(0) + 28 = 28
\]

Thus, the y-intercept is:

\[
\text{y-intercept: } (0, 28)
\]

### Summary of Key Points

- **Vertex:** \( (8, -36) \)
- **Smaller x-intercept:** \( (2, 0) \)
- **Larger x-intercept:** \( (14, 0) \)
- **y-intercept:** \( (0, 28) \)