Question

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

x	y
1	-3
2	-9
3	-27
4	-81

linear
quadratic
exponential

1. each of these functions grows as x gets larger and larger. which function eventually exceeds the others?

$f(x)=4.2x^{2}-1$
$g(x)=5.6x + 2$
$h(x)=2.5^{x}-6$
submit

Explanation:

First Problem: Classify the function from the table

Step1: Check linear (constant Δy)

Calculate differences in y-values:
$\Delta y_1 = -3 - (-1) = -2$
$\Delta y_2 = -9 - (-3) = -6$
$\Delta y_3 = -27 - (-9) = -18$
$\Delta y_4 = -81 - (-27) = -54$
Differences are not constant, so not linear.

Step2: Check quadratic (constant 2nd Δy)

Calculate differences of the differences:
$\Delta^2 y_1 = -6 - (-2) = -4$
$\Delta^2 y_2 = -18 - (-6) = -12$
$\Delta^2 y_3 = -54 - (-18) = -36$
Second differences are not constant, so not quadratic.

Step3: Check exponential (constant ratio)

Calculate ratios of consecutive y-values:
$\frac{-3}{-1} = 3$
$\frac{-9}{-3} = 3$
$\frac{-27}{-9} = 3$
$\frac{-81}{-27} = 3$
Ratio is constant, so it is exponential.

Step1: Classify each function type

• $g(x)=5.6x+2$: linear (degree 1)
• $f(x)=4.2x^2-1$: quadratic (degree 2)
• $h(x)=2.5^x-6$: exponential (base >1)

Step2: Compare growth rates

For large positive $x$, exponential functions grow faster than polynomial functions (linear/quadratic), as their growth is multiplicative rather than additive/polynomial.

Answer:

exponential

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Second Problem: Identify the fastest-growing function