Testinar Grade 8 math

Comparing Two Functions

Learn to reason about and solve Grade 8 problems involving Comparing Two Functions.

CH04-03 Functions and Linear Relationships

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Directions Solve each problem. Show your reasoning on the page or on separate paper. Check the answer key only after finishing.

Function $f$ is given by $f(x)=4x+1$. Function $g$ is given by the table: $g(0)=2,\ g(1)=5,\ g(2)=8$. Which function has the greater rate of change?
1. $f(x)$
2. $g(x)$
3. Both are equal
4. Cannot be determined
Function $h(x)=-2x+7$ and function $k$ shown in the table: $(0,7)$, $(1,5)$, $(2,3)$. Which statement is true?
1. $h$ has a steeper slope than $k$.
2. $k$ has a steeper slope than $h$.
3. Both have equal slopes.
4. The slopes cannot be compared.
Function $p(x)=\frac{3}{2}x-4$ and the table for function $q$: $(0,2)$, $(2,5)$, $(4,8)$. Which has a steeper slope?
1. $p(x)$
2. $q(x)$
3. Both have equal slopes.
4. Cannot be determined.
Plan A costs $$15$ per month: $A(m)=15+0.10m$. Plan B shown in table: $(0,$20)$, $(100,$25)$, $(200,$30)$, $(300,$35)$. Which plan has the higher base cost?
1. Plan A
2. Plan B
3. Both cost the same.
4. Cannot be determined.
Table: $(1,2)$, $(2,6)$, $(3,10)$; and $z(x)=3x+1$. Which crosses the $y$-axis higher?
1. Table function
2. $z(x)$
3. Both have same $y$-intercept.
4. Cannot be determined.
Function $A(t)=60t$ (mph). Function $B$ passes through $(0,1)$ and $(5,5)$ (on distance-time graph, with $t$ in hours). Which travels faster?
1. Function $A$
2. Function $B$
3. Both same speed.
4. Cannot be determined.
$S(m)=500+25m$ and table $(0,$600)$, $(3,$675)$, $(6,$750)$. Which starts with more money?
1. $S$
2. Table function
3. Both same.
4. Cannot be determined.
Table 1: $(0,1)$, $(2,5)$, $(4,9)$. Table 2: $(0,-2)$, $(1,-1)$, $(2,0)$. Which table represents a function with steeper rate of change?
1. Table 1
2. Table 2
3. Both have same rate.
4. Cannot be determined.

Teacher and parent reference

Answer key and explanations

A. $f(x)$
The slope of $f$ is $4$. For $g$: $\Delta y/\Delta x=3/1=3$. Since $4>3$, $f$ has the greater rate of change.
C. Both have equal slopes.
Slope of $h$: $-2$. From $k$'s table: $(5-7)/(1-0)=-2$. Both have slope $-2$.
C. Both have equal slopes.
Slope of $p$: $\frac{3}{2}=1.5$. From table $q$: $(5-2)/(2-0)=3/2=1.5$. Both have slope $1.5$.
B. Plan B
Plan A base cost: $$15$. Plan B at $m=0$: $$20$. Since $20>15$, Plan B is higher.
B. $z(x)$
Table slope: $(6-2)/(2-1)=4$. From $(1,2)$: $2=4(1)+b \Rightarrow b=-2$. So $y$-intercept is $-2$. For $z$: $y$-intercept is $1$. Since $1>-2$, $z$ is higher.
A. Function $A$
Speed is the slope. $A$ has slope $60$ mph. $B$ has slope $(5-1)/(5-0)=4/5=0.8$ mph. Since $60>0.8$, $A$ travels faster.
B. Table function
$S(0)=500$. Table shows $600$ at $m=0$. Since $600>500$, table function starts higher.
A. Table 1
Rate for Table 1: $(5-1)/(2-0)=2$. Rate for Table 2: $(-1-(-2))/(1-0)=1$. Since $2>1$, Table 1 is steeper.