Questions: 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 function's domain and range. f(x) = x^2 + x - 6

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 function's domain and range. f(x) = x^2 + x - 6
Transcript text: 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 function's domain and range. \[ f(x)=x^{2}+x-6 \]
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Solution

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Solution Steps

Step 1: Identify the quadratic function

The given quadratic function is \( f(x) = x^2 + x - 6 \).

Step 2: Find the vertex

To find the vertex of the quadratic function \( f(x) = ax^2 + bx + c \), use the formula \( x = -\frac{b}{2a} \). Here, \( a = 1 \), \( b = 1 \), and \( c = -6 \).

\[ x = -\frac{1}{2 \cdot 1} = -\frac{1}{2} = -0.5 \]

Now, substitute \( x = -0.5 \) back into the function to find the y-coordinate of the vertex.

\[ f(-0.5) = (-0.5)^2 + (-0.5) - 6 = 0.25 - 0.5 - 6 = -6.25 \]

So, the vertex is \( (-0.5, -6.25) \).

Step 3: Find the x-intercepts

To find the x-intercepts, set \( f(x) = 0 \) and solve for \( x \).

\[ x^2 + x - 6 = 0 \]

Factor the quadratic equation:

\[ (x + 3)(x - 2) = 0 \]

So, the x-intercepts are \( x = -3 \) and \( x = 2 \).

Final Answer

  • Vertex: \( (-0.5, -6.25) \)
  • X-intercepts: \( x = -3 \) and \( x = 2 \)
  • Axis of Symmetry: \( x = -0.5 \)
  • Domain: \( (-\infty, \infty) \)
  • Range: \( [-6.25, \infty) \)
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