Question

problem 12. consider the function $f(x) = 2x^3 + 4x^2 + 4x$. this question has three parts: (a) find the $x$-intercepts of $f(x)$. (b) find the $y$-intercept of $f(x)$. (c) describe the end behavior of the function using arrow notation.

Explanation:

Part (a)

Step1: Set \( f(x) = 0 \)

To find the \( x \)-intercepts, we set \( y = f(x) = 0 \), so we solve the equation \( 2x^3 + 4x^2 + 4x = 0 \).

Step2: Factor the equation

First, factor out the greatest common factor, which is \( 2x \):
\( 2x(x^2 + 2x + 2) = 0 \)
Now, we have two factors: \( 2x = 0 \) or \( x^2 + 2x + 2 = 0 \).

Step3: Solve \( 2x = 0 \)

For \( 2x = 0 \), dividing both sides by 2 gives \( x = 0 \).

Step4: Solve \( x^2 + 2x + 2 = 0 \)

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) for \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = 2 \), \( c = 2 \).
The discriminant \( D = b^2 - 4ac = 2^2 - 4(1)(2) = 4 - 8 = -4 \), which is negative. So, the quadratic equation \( x^2 + 2x + 2 = 0 \) has no real solutions.

Thus, the only real \( x \)-intercept is from \( 2x = 0 \), which is \( x = 0 \).

Step1: Set \( x = 0 \)

To find the \( y \)-intercept, we set \( x = 0 \) in the function \( f(x) \) and solve for \( y = f(0) \).

Step2: Substitute \( x = 0 \) into \( f(x) \)

\( f(0) = 2(0)^3 + 4(0)^2 + 4(0) = 0 + 0 + 0 = 0 \).

Step1: Analyze the leading term

The function is \( f(x) = 2x^3 + 4x^2 + 4x \). The leading term (the term with the highest power of \( x \)) is \( 2x^3 \).
The degree of the polynomial is 3 (odd), and the leading coefficient is \( 2 \) (positive).

Step2: Determine end behavior

For a polynomial with an odd degree and positive leading coefficient:

• As \( x \to +\infty \), the term \( 2x^3 \) dominates, and since \( x^3 \to +\infty \) when \( x \to +\infty \) and the coefficient is positive, \( f(x) \to +\infty \).
• As \( x \to -\infty \), \( x^3 \to -\infty \) (because an odd power of a negative number is negative), and with a positive coefficient, \( 2x^3 \to -\infty \), so \( f(x) \to -\infty \).

In arrow notation, this is written as:
As \( x \to +\infty \), \( f(x) \to +\infty \); as \( x \to -\infty \), \( f(x) \to -\infty \), or \( \lim_{x \to +\infty} f(x) = +\infty \) and \( \lim_{x \to -\infty} f(x) = -\infty \), which in arrow notation is \( x \to +\infty, f(x) \to +\infty \) and \( x \to -\infty, f(x) \to -\infty \).

Answer:

\( x = 0 \) (or the point \( (0, 0) \))

Part (b)