16.

Use the image to answer the questions.

(5 points)
Identify these key features of the parabola.

vertex: (
,
)

Smaller x-intercept : (
, 0)

larger x -intercept: (
, 0)

y-intercept: (0,
)

Question 2
17.

What are the key points on the graph of y = x2 − 16x + 28
?

(6 points)
Vertex: (
,
)

Smaller x-intercept: (
, 0)

Larger x-intercept (
, 0)

y- intercept: (0,
)

Axis of symmetry: x=

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1 answer

To find the key features of the parabola given by the equation \( y = x^2 - 16x + 28 \), we can follow these steps:

Step 1: Find the vertex

The vertex of a parabola in the standard form \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \).

For the equation \( y = x^2 - 16x + 28 \):

  • \( a = 1 \)
  • \( b = -16 \)
  • \( c = 28 \)

Calculate x-coordinate of the vertex: \[ x = -\frac{-16}{2 \cdot 1} = \frac{16}{2} = 8 \]

Calculate y-coordinate of the vertex: \[ y = (8)^2 - 16(8) + 28 = 64 - 128 + 28 = -36 \]

Vertex: \[ (8, -36) \]

Step 2: Find x-intercepts

To find the x-intercepts, we set \( y = 0 \): \[ 0 = x^2 - 16x + 28 \] We can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Plugging \( a = 1, b = -16, c = 28 \): \[ x = \frac{16 \pm \sqrt{(-16)^2 - 4 \cdot 1 \cdot 28}}{2 \cdot 1} \] \[ x = \frac{16 \pm \sqrt{256 - 112}}{2} \] \[ x = \frac{16 \pm \sqrt{144}}{2} \] \[ x = \frac{16 \pm 12}{2} \]

This gives us two solutions:

  1. \( x = \frac{28}{2} = 14 \) (larger x-intercept)
  2. \( x = \frac{4}{2} = 2 \) (smaller x-intercept)

X-intercepts:

  • Smaller x-intercept: \( (2, 0) \)
  • Larger x-intercept: \( (14, 0) \)

Step 3: Find y-intercept

To find the y-intercept, set \( x = 0 \): \[ y = 0^2 - 16(0) + 28 = 28 \] Y-intercept: \[ (0, 28) \]

Step 4: Axis of symmetry

The axis of symmetry is the x-coordinate of the vertex: \[ x = 8 \]

Summary of the Key Features:

  1. Vertex: \( (8, -36) \)
  2. Smaller x-intercept: \( (2, 0) \)
  3. Larger x-intercept: \( (14, 0) \)
  4. Y-intercept: \( (0, 28) \)
  5. Axis of symmetry: \( x = 8 \)

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