Question

15/40
match each polynomial function to its graph.
m(x)=x³+3x²+2x+1 a(x)=x²+14x+54 t(x)=-2x²-24x- h(x)=-x³+3x+

Explanation:

Step1: Analyze odd-degree polynomials

First, identify odd-degree functions: $m(x)=x^3+3x^2+2x+1$ (degree 3, leading coefficient positive) and $h(x)=-x^3+3x+...$ (degree 3, leading coefficient negative).

• For odd-degree polynomials:
• Positive leading coefficient: As $x\to+\infty$, $y\to+\infty$; as $x\to-\infty$, $y\to-\infty$. This matches the first leftmost graph (increasing right, decreasing left, single inflection).
• Negative leading coefficient: As $x\to+\infty$, $y\to-\infty$; as $x\to-\infty$, $y\to+\infty$. This matches the second graph from the left (decreasing right, increasing left, single inflection).

Step2: Analyze even-degree polynomials

Identify even-degree functions: $a(x)=x^2+14x+54$ (degree 2, leading coefficient positive) and $t(x)=-2x^2-24x-...$ (degree 2, leading coefficient negative).

• For even-degree polynomials:
• Positive leading coefficient: Parabola opens upward, U-shaped. This matches the third graph from the left.
• Negative leading coefficient: Parabola opens downward, ∩-shaped. This matches the rightmost graph.

Step3: Confirm matches

• $m(x)=x^3+3x^2+2x+1$ → Leftmost graph
• $h(x)=-x^3+3x+...$ → Second graph from left
• $a(x)=x^2+14x+54$ → Third graph from left
• $t(x)=-2x^2-24x-...$ → Rightmost graph

Answer:

1. $m(x)=x^3+3x^2+2x+1$ matches the leftmost cubic graph (rising right, falling left)
2. $h(x)=-x^3+3x+...$ matches the second cubic graph from the left (falling right, rising left)
3. $a(x)=x^2+14x+54$ matches the third upward-opening parabola graph
4. $t(x)=-2x^2-24x-...$ matches the rightmost downward-opening parabola graph