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Use the vertex and intercepts to sketch the graph of the quadratic function. Give the equation of the​ parabola's axis of symmetry. Use the graph to determine the domain and range of the function.

f(x) = (x+2)^2-9

Use the graphing tool to graph the function. Use the
vertex and one of the intercepts when drawing the
graph.

The axis of symmetry is blank.
(Type an equation. Simplify your answer.)
The domain of the function is blank.
(Type your answer in interval notation.)
The range of the function is blank.
(Type your answer in interval notation.)

1 answer

To analyze the quadratic function \( f(x) = (x+2)^2 - 9 \), we will identify the vertex, intercepts, axis of symmetry, domain, and range.

Step 1: Identify the Vertex

The function is in vertex form \( f(x) = a(x-h)^2 + k \), where the vertex is at the point \( (h, k) \).

For \( f(x) = (x+2)^2 - 9 \):

  • \( h = -2 \)
  • \( k = -9 \)

Thus, the vertex is at the point \( (-2, -9) \).

Step 2: Find the x-intercepts

To find the x-intercepts, set \( f(x) = 0 \): \[ 0 = (x+2)^2 - 9 \] Adding 9 to both sides: \[ 9 = (x+2)^2 \] Taking the square root of both sides: \[ \sqrt{9} = x + 2 \quad \text{or} \quad -\sqrt{9} = x + 2 \] This gives us two equations:

  1. \( x + 2 = 3 \) → \( x = 1 \)
  2. \( x + 2 = -3 \) → \( x = -5 \)

Thus, the x-intercepts are \( (1, 0) \) and \( (-5, 0) \).

Step 3: Find the y-intercept

To find the y-intercept, set \( x = 0 \): \[ f(0) = (0 + 2)^2 - 9 = 4 - 9 = -5 \] So, the y-intercept is \( (0, -5) \).

Step 4: Determine the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex. The equation of the axis of symmetry is given by \( x = h \): \[ \text{Axis of symmetry: } x = -2 \]

Step 5: Determine the Domain

The domain of a quadratic function is all real numbers: \[ \text{Domain: } (-\infty, \infty) \]

Step 6: Determine the Range

The range of the function is determined by the vertex. The vertex \( (-2, -9) \) is the lowest point of the parabola, and since it opens upwards, the range is: \[ \text{Range: } [-9, \infty) \]

Summary of Results

  1. The vertex of the parabola is \((-2, -9)\).
  2. The x-intercepts are \((1, 0)\) and \((-5, 0)\).
  3. The y-intercept is \((0, -5)\).
  4. The axis of symmetry is \( x = -2 \).
  5. The domain is \( (-\infty, \infty) \).
  6. The range is \( [-9, \infty) \).

Final Answers

  • The axis of symmetry is: \( x = -2 \)
  • The domain is: \( (-\infty, \infty) \)
  • The range is: \( [-9, \infty) \)