Questions: Analyze the polynomial function f(x)=(x+5)^2(x-6)^2 using parts (a) through (h) below. (b) Find the x-and y-intercepts of the graph of the function. The x-intercept(s) is/are -5,6. The y-intercept(s) is/are 900. (c) Determine the real zeros of the function and their multiplicity. Use this information to determine whether the graph crosses or touches the x-axis at each x-intercept. The real zero(s) of f is/are -5,6. The lesser zero is a zero of multiplicity 2, so the graph of f touches the x-axis at x=-5. The greater zero is a zero of multiplicity 2, so the graph of f touches the x-axis at x=6. (d) Use a graphing utility to graph the function. The graphs are shown in the viewing window Xmin=-14, Xmax=14, Xscl=2, Ymin=-1200, Ymax=1200, Yscl=120 Choose the correct graph below. A. B. C. D. (e) Approximate the turning points of the graph. The turning point(s) of the graph is/are

Analyze the polynomial function f(x)=(x+5)^2(x-6)^2 using parts (a) through (h) below. (b) Find the x-and y-intercepts of the graph of the function. The x-intercept(s) is/are -5,6. The y-intercept(s) is/are 900. (c) Determine the real zeros of the function and their multiplicity. Use this information to determine whether the graph crosses or touches the x-axis at each x-intercept. The real zero(s) of f is/are -5,6. The lesser zero is a zero of multiplicity 2, so the graph of f touches the x-axis at x=-5. The greater zero is a zero of multiplicity 2, so the graph of f touches the x-axis at x=6. (d) Use a graphing utility to graph the function. The graphs are shown in the viewing window Xmin=-14, Xmax=14, Xscl=2, Ymin=-1200, Ymax=1200, Yscl=120 Choose the correct graph below. A. B. C. D. (e) Approximate the turning points of the graph. The turning point(s) of the graph is/are
Transcript text: Analyze the polynomial function $f(x)=(x+5)^{2}(x-6)^{2}$ using parts (a) through (h) below. (b) Find the $x$-and $y$-intercepts of the graph of the function. The $x$-intercept(s) is/are $-5,6$. The $y$-intercept(s) is/are 900. (c) Determine the real zeros of the function and their multiplicity. Use this information to determine whether the graph crosses or touches the $x$-axis at each $x$-intercept. The real zero(s) of $f$ is/are $-5,6$. The lesser zero is a zero of multiplicity 2, so the graph of $f$ touches the $x$-axis at $x=-5$. The greater zero is a zero of multiplicity 2, so the graph of $f$ touches the $x$-axis at $x=6$. (d) Use a graphing utility to graph the function. The graphs are shown in the viewing window $X_{\min}=-14, X_{\max}=14, X_{\text{scl}}=2, Y_{\min}=-1200, Y_{\max}=1200, Y_{\text{scl}}=120$ Choose the correct graph below. A. B. C. D. (e) Approximate the turning points of the graph. The turning point(s) of the graph is/are $\square$
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Solution

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Solution Steps

Step 1: Find the x- and y-intercepts of the graph of the function

  • The x-intercept(s) is/are found by setting \( f(x) = 0 \). \[ f(x) = (x + 5)^2 (x - 6)^2 = 0 \] Solving for \( x \): \[ (x + 5)^2 = 0 \quad \text{or} \quad (x - 6)^2 = 0 \] \[ x = -5 \quad \text{or} \quad x = 6 \] Therefore, the x-intercepts are \( x = -5 \) and \( x = 6 \).

  • The y-intercept is found by setting \( x = 0 \). \[ f(0) = (0 + 5)^2 (0 - 6)^2 = 25 \cdot 36 = 900 \] Therefore, the y-intercept is \( y = 900 \).

Step 2: Determine the real zeros of the function and their multiplicity
  • The real zeros are \( x = -5 \) and \( x = 6 \).
  • The multiplicity of each zero is determined by the exponent of the factor in the polynomial. \[ (x + 5)^2 \quad \text{has multiplicity} \quad 2 \] \[ (x - 6)^2 \quad \text{has multiplicity} \quad 2 \] Therefore, the real zeros are \( x = -5 \) with multiplicity 2 and \( x = 6 \) with multiplicity 2.
Step 3: Choose the correct graph
  • Since both zeros have even multiplicities, the graph touches the x-axis at these points but does not cross it.
  • The correct graph is the one where the graph touches the x-axis at \( x = -5 \) and \( x = 6 \) without crossing it.

Final Answer

  • The x-intercepts are \( x = -5 \) and \( x = 6 \).
  • The y-intercept is \( y = 900 \).
  • The real zeros are \( x = -5 \) with multiplicity 2 and \( x = 6 \) with multiplicity 2.
  • The correct graph is the one where the graph touches the x-axis at \( x = -5 \) and \( x = 6 \) without crossing it.
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