Task Sets: Compare function properties - Common Core Math - HS Functions

Concepts

Function B is given by $g(x) = x^2 - 4x$ .

Evaluate $g(0)$ to find the y-intercept of Function B. Enter the exact value.

A student looks at the graph of Function A and concludes "the maximum
is exactly 4.5." A classmate says "the graph only supports an estimate
of approximately 4.5." Who is correct, and why?

The first student — any point visible on a graph is exact.

The classmate — graph readings are estimates unless the value falls exactly on a labeled gridline.

Both are correct — it depends on how carefully you read the graph.

Neither — you must use algebra to determine the maximum of any function.

Function A and Function B are two different functions. A student
observes that $f(3) > g(3)$ and concludes "Function A is always
greater than Function B." What is wrong with this reasoning?

Nothing — if one function is greater at one point, it is greater everywhere.

The comparison only shows that $f(3) > g(3)$ at a single input value.
The functions could cross or reverse order at other x-values.

The student should have compared $f(0)$ and $g(0)$ instead of at $x = 3$ .

The student needs to compare both functions at every integer to make a valid conclusion.

Procedures

Function A is given by the table below.

$x$	$f(x)$
0	$-3$
1	$0$
2	$1$
3	$0$

Function B is given by $g(x) = x^2 - 4x$ .

Which function has the greater y-intercept?

Function A, because $f(0) = -3$ .

Function B, because $g(0) = 0$ .

They are equal.

Function A, because it has more visible table entries near zero.

Function B is given by $h(x) = -2x^2 + 12x - 13$ .

Find the maximum value of Function B. Enter the exact value.

Function A (verbal description): "A car accelerates from 0 mph to
60 mph over an 8-second interval."

Function B (table):

$t$ (seconds)	speed (mph)
0	0
8	50

What is the average rate of change (average acceleration) of Function A
over the interval $[0, 8]$ ? Enter your answer in mph per second.

Using the same functions from procedures-3:

Function A: car goes from 0 to 60 mph in 8 seconds (average rate = 7.5 mph/s).
Function B: car goes from 0 to 50 mph in 8 seconds.

What is the average rate of change (average acceleration) of Function B
over $[0, 8]$ ? Enter your answer in mph per second.

Function A (algebraic): $f(x) = -x^2 + 6x - 5$

Function B (graph): a downward-opening parabola with vertex at
approximately $(1, 4)$ .

A student completes the square on Function A and finds that its
maximum is exactly 4, occurring at $x = 3$ . Based on this and the
graph description, which statement is best supported?

Function A has a larger maximum because algebra gives exact values while the graph is an estimate.

Function B has a larger maximum because its vertex appears to be at 4 from the graph.

Both functions appear to have a maximum of 4; Function A''s maximum
is confirmed exactly as 4, while Function B''s maximum is estimated
as approximately 4 from the graph.

Function B has a larger maximum because graphical methods are more reliable than algebra.

HSF.IF.C.9 Tutor Intake — Comparing Properties Across Representations

Concepts

Procedures