Common Core: High School - Functions - Exponential Functions Exceeding Polynomial Functions: CCSS.Math.Content.HSF-LE.A.3

1

Which value for proves that the function will increases faster than the function ?

$x\geq2.6$

$x\geq1$

$x\geq2$

$x\geq6.2$

$x\geq0$

Explanation

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

For the purpose of Common Core Standards, "Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function" falls within the Cluster A of "Construct and compare linear, quadratic, and exponential models and solve problems" concept (CCSS.MATH.CONTENT.HSF-LE.A.3).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph .

Screen shot 2016 02 09 at 9.53.13 am

Step 2: Use technology to graph .

Q2 2

Step 3: Compare the graphs of and .

Screen shot 2016 02 09 at 9.53.25 am

Graphically, it appears that is the point where increases more rapidly than . Substitute this value into both functions to algebraic verify the assumption.

$f(2.6)=e^{2.6}=(2.718)^{2.6}=13.464$

Since

$13.464>13\Rightarrow f(2.6)>g(2.6)\Rightarrow x\geq2.6$

Step 4: Answer the question.

For values $x\geq2.6$ , $f(x)\geq g(x)$ .

2

Which value for proves that the function will increases faster than the function ?

$x\geq2$

$x\geq1$

$x\geq0$

$x\geq3$

$x\geq1.5$

Explanation

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

For the purpose of Common Core Standards, "Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function" falls within the Cluster A of "Construct and compare linear, quadratic, and exponential models and solve problems" concept (CCSS.MATH.CONTENT.HSF-LE.A.3).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph .

Q12

Step 2: Use technology to graph .

Q2 2

Step 3: Compare the graphs of and .

Q12 3

Graphically, it appears that is the point where increases more rapidly than . Substitute this value into both functions to algebraic verify the assumption.

$f(2)=e^{2}=(2.718)^{2}=7.389$

Since

$7.389>7\Rightarrow f(2)>g(2)\Rightarrow x\geq2$

Step 4: Answer the question.

For values $x\geq2$ , $f(x)\geq g(x)$ .

3

Which value for proves that the function will increases faster than the function ?

$x\geq2.9$

$x\leq2.8$

$x\geq2$

$x\geq3.12$

$x\geq1$

Explanation

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

For the purpose of Common Core Standards, "Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function" falls within the Cluster A of "Construct and compare linear, quadratic, and exponential models and solve problems" concept (CCSS.MATH.CONTENT.HSF-LE.A.3).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph .

Q11

Step 2: Use technology to graph .

Q2 2

Step 3: Compare the graphs of and .

Q11 3

Graphically, it appears that is the point where increases more rapidly than . Substitute this value into both functions to algebraic verify the assumption.

$f(2.9)=e^{2.9}=(2.718)^{2.9}=18.174$

Since

$18.174>17.4\Rightarrow f(2.9)>g(2.9)\Rightarrow x\geq2.9$

Step 4: Answer the question.

For values $x\geq2.9$ , $f(x)\geq g(x)$ .

4

Which value for proves that the function will increases faster than the function ?

$x\geq2.68$

$x\geq2$

$x\geq3.45$

$x\geq1$

$x\geq3.5$

Explanation

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

For the purpose of Common Core Standards, "Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function" falls within the Cluster A of "Construct and compare linear, quadratic, and exponential models and solve problems" concept (CCSS.MATH.CONTENT.HSF-LE.A.3).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph .

Q10

Step 2: Use technology to graph .

Q2 2

Step 3: Compare the graphs of and .

Q10 3

Graphically, it appears that is the point where increases more rapidly than . Substitute this value into both functions to algebraic verify the assumption.

$f(2.68)=e^{2.68}=(2.718)^{2.68}=14.585$

Since

$14.585>14.4\Rightarrow f(2.68)>g(2.68)\Rightarrow x\geq2.68$

Step 4: Answer the question.

For values $x\geq2.68$ , $f(x)\geq g(x)$ .

5

Which value for proves that the function will increases faster than the function ?

$\text{ For all values of }x\text{, } f(x)=e^x \text{ is larger than } y=x+1.$

$x\geq1$

$x\geq-1$

$x\geq0$

$x\geq2$

Explanation

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

For the purpose of Common Core Standards, "Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function" falls within the Cluster A of "Construct and compare linear, quadratic, and exponential models and solve problems" concept (CCSS.MATH.CONTENT.HSF-LE.A.3).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph .

Step 2: Use technology to graph .

Q2 2

Step 3: Compare the graphs of and .

Q4 3

Graphically, it appears that is larger than for all values of .

6

Which value for proves that the function will increases faster than the function ?

$\text{ For all values of }x\text{, } f(x)=e^x \text{ is larger than } y=x.$

$x\geq2$

$x\geq0$

$x\geq1$

$x\geq8$

Explanation

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

For the purpose of Common Core Standards, "Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function" falls within the Cluster A of "Construct and compare linear, quadratic, and exponential models and solve problems" concept (CCSS.MATH.CONTENT.HSF-LE.A.3).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph .

Step 2: Use technology to graph .

Q2 2

Step 3: Compare the graphs of and .

Q2 3

Graphically, it appears that is larger than for all values of .

7

Which value for proves that the function will increases faster than the function ?

$x\geq1.36$

$x\geq 2$

$x\geq1.6$

$x\geq1$

$x\geq0$

Explanation

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

For the purpose of Common Core Standards, "Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function" falls within the Cluster A of "Construct and compare linear, quadratic, and exponential models and solve problems" concept (CCSS.MATH.CONTENT.HSF-LE.A.3).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph .

Step 2: Use technology to graph .

Q2 2

Step 3: Compare the graphs of and .

Q3 3

Graphically, it appears that is the point where increases more rapidly than . Substitute this value into both functions to algebraic verify the assumption.

$f(1.36)=e^{1.36}=(2.718)^{1.36}=3.896$

Since

$3.896>3.72\Rightarrow f(1.36)>g(1.36)\Rightarrow x\geq1.36$

Step 4: Answer the question.

For values $x\geq1.36$ , $f(x)\geq g(x)$ .

8

Which value for proves that the function will increases faster than the function ?

$x\geq2.2$

$x\geq1.2$

$x\geq3.1$

$x\geq0$

$x\geq-1$

Explanation

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

For the purpose of Common Core Standards, "Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function" falls within the Cluster A of "Construct and compare linear, quadratic, and exponential models and solve problems" concept (CCSS.MATH.CONTENT.HSF-LE.A.3).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph .

Step 2: Use technology to graph .

Q2 2

Step 3: Compare the graphs of and .

Q7 3

Graphically, it appears that is the point where increases more rapidly than . Substitute this value into both functions to algebraic verify the assumption.

$f(2.2)=e^{2.2}=(2.718)^{2.2}=9.025$

Since

$9.025>8.4\Rightarrow f(2.2)>g(2.2)\Rightarrow x\geq2.2$

Step 4: Answer the question.

For values $x\geq2.2$ , $f(x)\geq g(x)$ .

9

Which value for proves that the function will increases faster than the function ?

$x\geq2.7$

$x\geq2$

$x\geq1$

$x\geq3$

$x\geq0$

Explanation

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

For the purpose of Common Core Standards, "Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function" falls within the Cluster A of "Construct and compare linear, quadratic, and exponential models and solve problems" concept (CCSS.MATH.CONTENT.HSF-LE.A.3).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph .

Step 2: Use technology to graph .

Q2 2

Step 3: Compare the graphs of and .

Q8 3

Graphically, it appears that is the point where increases more rapidly than . Substitute this value into both functions to algebraic verify the assumption.

$f(2.5)=e^{2.7}=(2.718)^{2.7}=14.880$

Since

$14.880>14.1\Rightarrow f(2.7)>g(2.7)\Rightarrow x\geq2.7$

Step 4: Answer the question.

For values $x\geq2.7$ , $f(x)\geq g(x)$ .

10

Which value for proves that the function will increases faster than the function ?

$x\geq1.51$

$x\geq 2$

$x\geq1$

$x\geq0$

$x\geq2.5$

Explanation

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

For the purpose of Common Core Standards, "Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function" falls within the Cluster A of "Construct and compare linear, quadratic, and exponential models and solve problems" concept (CCSS.MATH.CONTENT.HSF-LE.A.3).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph .

Step 2: Use technology to graph .

Q2 2

Step 3: Compare the graphs of and .

Q6 3

Graphically, it appears that is the point where increases more rapidly than . Substitute this value into both functions to algebraic verify the assumption.