Question

1. both of these functions grow as x gets larger and larger. which function eventually exceeds the other?

$f(x) = 5x^{2} + x$
$g(x) = 5x + \frac{9}{2}$

1. is this function linear, quadratic, or exponential?

x	y
-9	84.9
-8	76.4
-7	67.9
-6	59.4
-5	50.9

options: linear, quadratic, exponential

Explanation:

First Problem:

Step1: Compare function degrees

$f(x)=5x^2+x$ (degree 2, quadratic), $g(x)=5x+\frac{9}{2}$ (degree 1, linear)

Step2: Analyze growth as $x\to\infty$

Quadratic functions grow faster than linear functions for large $x$.

Step1: Check first differences of $y$

Calculate $\Delta y$ between consecutive points:
$84.9-76.4=8.5$, $76.4-67.9=8.5$, $67.9-59.4=8.5$, $59.4-50.9=8.5$

Step2: Classify by constant difference

Linear functions have constant first differences of $y$ for constant $\Delta x$.

Answer:

$f(x) = 5x^2 + x$ eventually exceeds $g(x) = 5x + \frac{9}{2}$ as $x$ gets larger.

---

Second Problem: