Question

describe the end behavior of the quadratic function shown below. pick all that apply.
$f(x) = 2x^2 + 3x + 5$
show your work here
hint: to add infinity (∞), type \infinity\

the function values increase as $x$ approaches positive infinity.
the function values decrease as $x$ approaches positive infinity.
the function values increase as $x$ approaches negative infinity.
the function values decrease as $x$ approaches negative infinity.

Explanation:

Step1: Analyze the leading coefficient

For a quadratic function \( f(x) = ax^2 + bx + c \), the leading coefficient is \( a \). Here, \( a = 2 \), which is positive.

Step2: Determine the end behavior

A quadratic function with a positive leading coefficient has a parabola that opens upward. So, as \( x \to +\infty \), \( f(x) \to +\infty \) (function values increase), and as \( x \to -\infty \), \( f(x) \to +\infty \) (function values increase). Wait, no—wait, when the parabola opens upward, the left end (as \( x \to -\infty \)): let's think about the graph. For \( y = ax^2 + bx + c \) with \( a>0 \), as \( x \) becomes very negative (approaches \( -\infty \)), \( x^2 \) is positive and large, so \( f(x) \) increases. As \( x \) becomes very positive (approaches \( +\infty \)), \( x^2 \) is positive and large, so \( f(x) \) increases. Wait, but let's check the options. The options are:

1. The function values increase as \( x \) approaches positive infinity. (True)
2. The function values decrease as \( x \) approaches positive infinity. (False)
3. The function values increase as \( x \) approaches negative infinity. (True)
4. The function values decrease as \( x \) approaches negative infinity. (False)

Wait, no—wait, maybe I made a mistake. Wait, for a parabola opening upward ( \( a>0 \) ), the vertex is the minimum point. So, as \( x \) moves from left to right, before the vertex, the function decreases, after the vertex, it increases. But for end behavior (as \( x \to \pm\infty \)), we look at the leading term. The leading term is \( 2x^2 \). As \( x \to +\infty \), \( 2x^2 \to +\infty \), so \( f(x) \to +\infty \) (increase). As \( x \to -\infty \), \( x^2 \) is positive (since squaring a negative gives positive), so \( 2x^2 \to +\infty \), so \( f(x) \to +\infty \) (increase). Wait, but that would mean both "increase as \( x \to +\infty \)" and "increase as \( x \to -\infty \)" are true? But let's check the options again.

Wait, the function is \( f(x) = 2x^2 + 3x + 5 \). The leading term is \( 2x^2 \), which dominates as \( x \) is very large in magnitude (positive or negative). So:

- As \( x \to +\infty \): \( 2x^2 \) is positive and grows, so \( f(x) \) increases. So option 1 is correct.
- As \( x \to -\infty \): \( x^2 \) is positive (since \( (-\infty)^2 = +\infty \)), so \( 2x^2 \) grows, so \( f(x) \) increases. So option 3 is correct? Wait, but that seems conflicting with my initial thought. Wait, no—let's take a concrete example. Let \( x = -100 \): \( f(-100) = 2(10000) + 3(-100) + 5 = 20000 - 300 + 5 = 19705 \). \( x = -200 \): \( f(-200) = 2(40000) + 3(-200) + 5 = 80000 - 600 + 5 = 79405 \). So as \( x \) goes from -200 to -100 (moving towards positive, but still negative), \( f(x) \) increases. So as \( x \to -\infty \), \( f(x) \) increases. As \( x \to +\infty \), \( x = 100 \): \( f(100) = 2(10000) + 3(100) + 5 = 20000 + 300 + 5 = 20305 \). \( x = 200 \): \( f(200) = 2(40000) + 3(200) + 5 = 80000 + 600 + 5 = 80605 \). So as \( x \to +\infty \), \( f(x) \) increases. So the correct options are:

- The function values increase as \( x \) approaches positive infinity.
- The function values increase as \( x \) approaches negative infinity.

Wait, but let's check the options given. The options are:

1. The function values increase as \( x \) approaches positive infinity. (True)
2. The function values decrease as \( x \) approaches positive infinity. (False)
3. The function values increase as \( x \) approaches negative infinity. (True)
4. The function values decrease as \( x \) approaches negative infinity. (False)

So the correct ones are the first and third options.

Answer:

The correct options are:

• The function values increase as \( x \) approaches positive infinity.
• The function values increase as \( x \) approaches negative infinity.

(In the format of the options, if we list them with their identifiers:
A. The function values increase as \( x \) approaches positive infinity.
C. The function values increase as \( x \) approaches negative infinity.

Assuming the options are labeled A, B, C, D as:
A. The function values increase as \( x \) approaches positive infinity.
B. The function values decrease as \( x \) approaches positive infinity.
C. The function values increase as \( x \) approaches negative infinity.
D. The function values decrease as \( x \) approaches negative infinity.

Then the answer is A and C.)