9th grade · Math · Unit 19 of 22
Comparing Functions
Key features and growth
What your child practices
Reading key features of a function from a formula, a table or a graph (highest point, when it rises, intercepts), comparing two functions shown in different ways, and seeing that exponential growth eventually passes linear and quadratic growth.
New York State Learning Standards learning standard: AI-F.IF.4; AI-F.IF.9; AI-F.LE.3 · interpreting key features of functions in context, comparing functions shown in different ways, and seeing that exponential growth eventually exceeds linear and quadratic growth
How we explain it
Children can open this short lesson before they practice.
- Key features: intercepts, where a function rises or falls, and its greatest or least value.
- To compare two functions shown in different ways, find the same feature in each, such as the rate of change.
- Exponential growth, like 2ˣ, eventually passes any linear or quadratic growth.
Example: f(x) = 3x + 1. g has values 2, 7, 12 at x = 0, 1, 2. Which grows faster? f grows by 3 for each 1 in x. g grows by 5 for each 1 in x. Answer: g
Comparing Functions in New York
In the New York State Learning Standards, this is unit 19 of 22 in 9th grade Math, after Graphs of Equations & Inequalities and before Parabolas & Lines.
Gradelings describes a child's practice on this unit as Beginning, Approaching, Meeting or Exceeding (Four practice steps). That reflects practice, not a report-card mark: the teacher decides the mark.
Sample questions
A taste of this unit. In the app, questions change every time and get easier or harder to match your child.
Question 1
A ball is kicked from the ground. Its height is h(t) = −16t² + 32t feet after t seconds. When is it highest, in seconds?
Height of a ball t (seconds) h (feet) 0 0 1 16 2 0 - 0
- 3
- 1
- 2
1
Hint: The heights rise, then fall. Find the time with the greatest height; the table is symmetric about it.
Question 2
A ball is kicked from the ground. Its height is h(t) = −16t² + 64t feet after t seconds. When is it highest, in seconds?
Height of a ball t (seconds) h (feet) 0 0 1 48 2 64 3 48 4 0 - 3
- 0
- 2
- 4
2
Hint: The heights rise, then fall. Find the time with the greatest height; the table is symmetric about it.
Question 3
f(x) = 2x + 8. Compare it with g, which starts at 1 and goes up 6 for each 1 in x. Which has the greater y-intercept?
- f
- they are the same
- g
f
Hint: The y-intercept is the value at x = 0: f(0) = 8 and g(0) = 1.
Question 4
f(x) = 2x + 8. Compare it with g (in the table). Which has the greater rate of change?
g(x) x g(x) 0 1 1 6 2 11 3 16 - g
- they are the same
- f
g
Hint: f's rate is its slope, 2. g goes up 5 for each 1 in x.
Question 5
f(x) = 20x and g(x) = 2ˣ. Starting at x = 2, what is the first whole number x where g(x) is greater than f(x)?
x f(x) g(x) 6 120 64 7 140 128 8 160 256 - 9
- 10
- 7
- 8
8
Hint: Make a table: compare 2ˣ with the linear values for x = 2, 3, 4, …
Question 6
f(x) = 100x and g(x) = 2ˣ. For very large x, which is greater?
- g(x) = 2ˣ
- f(x) = 100x
- they stay equal
g(x) = 2ˣ
Hint: A quantity that doubles each step eventually passes any linear or quadratic one.