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Let's go through the questions step-by-step based on the function \( f(x) = 2x^2 + 20x + 48 \):

### a. What is the function's leading term?
- The leading term of a polynomial is the term with the highest power of \(x\). For \( f(x) = 2x^2 + 20x + 48 \), the leading term is **\( 2x^2 \)**.

### b. What is the function's degree?
- The degree of a polynomial is the highest exponent of \(x\). Here, the degree is **2**, since the highest power of \(x\) in the function is \(x^2\).

### c. What is the function's end behavior?
- End behavior refers to how the function behaves as \(x\) approaches infinity (\( x \to \infty \)) or negative infinity (\( x \to -\infty \)).
    - As \( x \to \infty \), the leading term \(2x^2\) dominates, and since \(2x^2\) is positive, \( f(x) \to \infty \).
    - As \( x \to -\infty \), the leading term \(2x^2\) still dominates, and because it's positive, \( f(x) \to \infty \).

So:
- As \( x \to \infty \), \( f(x) \to \infty \).
- As \( x \to -\infty \), \( f(x) \to \infty \).

### d. What is the vertical intercept for \(f\)?
- The vertical intercept occurs when \(x = 0\). Substituting \(x = 0\) into the function:
  \[
  f(0) = 2(0)^2 + 20(0) + 48 = 48
  \]
  So, the vertical intercept is **48**.

### e. Write \( f(x) = 2x^2 + 20x + 48 \) in factored form.
- First, factor out the common factor of 2:
  \[
  f(x) = 2(x^2 + 10x + 24)
  \]
- Now factor \(x^2 + 10x + 24\):
  \[
  x^2 + 10x + 24 = (x + 4)(x + 6)
  \]
- So the factored form is:
  \[
  f(x) = 2(x + 4)(x + 6)
  \]

### f. Determine the function's roots.
- From the factored form \(f(x) = 2(x + 4)(x + 6)\), the roots are:
  \[
  x + 4 = 0 \quad \text{or} \quad x + 6 = 0
  \]
  Solving these gives \(x = -4\) and \(x = -6\).
- So the roots are **-4, -6**.

### g. Use a graphing calculator to graph \(f\) and check your work.
- You can use any graphing tool or calculator to visualize the graph. Based on the roots and intercept, the graph is a parabola that opens upwards, with the roots at \(x = -4\) and \(x = -6\), and a vertical intercept at \(y = 48\).

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Would you like more details or have any further questions on these steps? Here are some related questions to extend your learning:

1. How does the leading coefficient affect the end behavior of the graph?
2. How can you find the axis of symmetry for the quadratic function?
3. How would the graph change if the leading coefficient were negative?
4. What is the vertex of this quadratic function?
5. Can you determine if this function has a maximum or minimum point?

**Tip:** For quadratic functions in the form \( ax^2 + bx + c \), the vertex can be found using the formula \( x = -\frac{b}{2a} \).

Suppose that f(x) = 2x^2 + 20x + 48. Answer the following questions: a. What is the function's leading term? b. What is the function's degree? c. What is the function's end behavior? d. What is the vertical intercept for f? e. Write f(x) in factored form. f. Determine the function's roots.

This problem involves analyzing the quadratic function f(x) = 2x^2 + 20x + 48. It covers finding the leading term, degree, factored form, roots, end behavior, and vertical intercept. Suitable for Grades 9-11.

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