Common Core Standard HSF-LE.A.1 Questions

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Grade 10 Functions and Relations CCSS: HSF-LE.A.1, HSF-LE.A.1a

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What is the value of

$Delta f = f(x + Delta x) - f(x)$ , using the definition of

$f(x) ?$ Simplify your answer fully.

Grade 10 Exponents CCSS: HSF-LE.A.1, HSF-LE.A.1c

Grade 10 Functions and Relations CCSS: HSF-LE.A.1, HSF-LE.A.1a

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Because the resulting equation for $Delta f$ is also linear, it will increase at a constant rate.
Because there is no slope in the resulting equation, the value of $Delta f$ is constant.
Since the resulting equation for $Delta f$ has no $b$ value, it is independent of the y-intercept. As such, $Delta f$ will increase by equal amounts over equal intervals.
Since $Delta f$ is dependent only on the length of the interval, as long as the interval $Delta x$ is constant, $Delta f$ will be the same.

Grade 10 Exponents CCSS: HSF-LE.A.1, HSF-LE.A.1c

Grade 10 Functions and Relations CCSS: HSF-LE.A.1, HSF-LE.A.1a

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What if, for the linear function

$f(x)$ ,

$m =0 ?$

This means that the above reasoning is invalid.
For how the function is defined above, $m$ cannot equal zero.
It makes things easier, since the growth rate of $f(x)$ simply becomes zero (and thus constant) for all intervals.
It does not change anything, since $Delta f$ is not dependent on $m$ .

Grade 10 Exponents CCSS: HSF-LE.A.1, HSF-LE.A.1c

$\ \ \ \ \ \ \ \ \ \ \ mathbf{"Number of Days on the Job"} \ \ \ \ \ \ \ \ \ \ \$	$\ \ \ \ \ \ \ \ \ \ \mathbf{"Number of Sales"} \ \ \ \ \ \ \ \ \ \$
$1$	$1$
$2$	$2$
$3$	$4$
$4$	$7$
$5$	$10$
$6$	$14$