Task Sets: Prove growth properties of functions - Common Core Math - HS Functions

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For $f(x) = 5x + 2$ , what is $f(x + 3)$ ?

$5x + 3 + 2 = 5x + 5$

$5(x + 3) + 2 = 5x + 17$

$5x + 2 + 3 = 5x + 5$

$(5x + 2) \cdot 3 = 15x + 6$

Which exponent rule is used to rewrite $b^{x+d}$ ?

$b^{x+d} = b^x + b^d$

$b^{x+d} = b^x \cdot b^d$

$b^{x+d} = (b+b)^{x+d} = (2b)^{x+d}$

$b^{x+d} = b^{xd}$

A linear function $f(x) = mx + b$ has slope $m$ . What does slope represent?

The starting value when $x = 0$ .

The constant rate of change — how much $f$ increases per unit increase in $x$ .

The $y$ -value at the maximum.

The ratio of consecutive outputs.

For $f(x) = 4x + 7$ , use the formula $f(x + d) - f(x) = md$ with $m = 4$ and $d = 5$ . What is the constant difference $f(x + 5) - f(x)$ ?

For $f(x) = -3x + 10$ , which expression correctly shows $f(x + 2) - f(x)$ ?

$-3(x + 2) + 10 - (-3x + 10) = -6$

$-3x + 2 + 10 - (-3x + 10) = 2$

$(-3x + 10) \cdot 2 = -6x + 20$

$-3(x + 2) - (-3x) = -6x - 6$

For $g(x) = 3 \cdot 2^x$ , use the formula $g(x + d)/g(x) = b^d$ with $b = 2$ and $d = 4$ . What is the constant ratio $g(x + 4)/g(x)$ ?

For $g(x) = 5 \cdot 3^x$ , which expression correctly computes $g(x + 2) / g(x)$ ?

$\frac{5 \cdot 3^{x+2}}{5 \cdot 3^x} = \frac{3^x \cdot 3^2}{3^x} = 9$

$\frac{5 \cdot 3^x + 5 \cdot 3^2}{5 \cdot 3^x} = 1 + \frac{9}{3^x}$ , which depends on $x$

$\frac{5 \cdot 3^{2x}}{5 \cdot 3^x} = 3^x$ , which depends on $x$

$\frac{5 \cdot 3^{x+2}}{5 \cdot 3^x} = x + 2$

For a linear function $f(x) = mx + b$ , the proof shows that $f(x + d) - f(x) = md$ . What does this result confirm about the slope $m$ ?

The slope $m$ equals the constant difference divided by the interval width $d$ .

The slope $m$ equals the constant difference (the result $md$ itself).

The slope $m$ is irrelevant to the constant difference.

The slope $m$ equals the interval width $d$ .

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