Graphs of Polynomial Functions - Pre-Calculus

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Question

Give the -intercept of the graph of the function

$f(x) = 5 \cdot 4^{x- 3}- 3$

Round to the nearest tenth, if applicable.

Answer

The -intercept is , where :

$f(x) = 5 \cdot 4^{x- 3}- 3$

$5 \cdot 4^{a- 3}- 3 = 0$

$5 \cdot 4^{a- 3}- 3 + 3= 0 + 3$

$5 \cdot 4^{a- 3}= 3$

$5 \cdot 4^{a- 3} \div 5= 3 \div 5$

$4^{a- 3} = 0.6$

$\ln \left (4^{a- 3} \right )=\ln 0.6$

$(a- 3)\ln 4 =\ln 0.6$

$a- 3=\frac{\ln 0.6}{\ln 4 } \approx \frac{-0.5108}{1.3863}\approx -0.3685$

$a- 3+3 \approx -0.3685 + 3$

$a \approx 2.63$

The -intercept is .

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Question

Which of the following is an accurate graph of $f(x) = -(x + 4)^{2}$ ?

Answer

is a parabola, because of the general $f(x) = x^{2}$ structure. The parabola opens downward because .

Solving $-(x + 4)^{2} = 0$ tells the x-value of the x-axis intercept;

$0 = (x + 4)^{2}$

$\sqrt{0} = \sqrt{(x + 4)^{2}}$

The resulting x-axis intercept is: .

Setting tells the y-value of the y-axis intercept;

$y = -(0 + 4)^{2}$

$= -1(4)^{2}$

The resulting y-axis intercept is:

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Question

Graph the following function and identify the zeros.

Answer

This question tests one's ability to graph a polynomial function.

For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior" falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).

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 algebraic technique to factor the function.

Separating the function into two parts...

Factoring a negative one from the second set results in...

Factoring out from the first set results in...

The new factored form of the function is,

.

Now, recognize that the first binomial is a perfect square for which the following formula can be used

$(x^2-b^2)=(x-\sqrt{b})(x+\sqrt{b})$

since

$b=1\Rightarrow \sqrt{b}=\pm1$

thus the simplified, factored form is,

.

Step 2: Identify the roots of the function.

To find the roots of a function set its factored form equal to zero and solve for the possible x values.

$\(x-1)(x+1)(x-2)=0 \x-1=0\Rightarrow x=1 \x+1=0\Rightarrow x=-1 \x-2=0\Rightarrow x=2$

Step 3: Create a table of pairs.

$\begin{tabular}{ |c|c|c|c|c|c|c| } \hline x & f(x) & y \ \hline -2& f(-2) & -12 \ -1 & f(-1) & 0 \ 0&f(0)& 2\ 1.5&f(1.5)&-0.625\2&f(2)&0 \ 3&f(3)&8 \\hline \end{tabular}$

The values in the table are found by substituting in the x values into the function as follows.

$\f(-2)=(-2)^3-2(-2)^2-(-2)+2=-8-8+2+2=-16+4=-12 \f(-1)=(-1)^3-2(-1)^2-(-1)+2=-1-2+1+2=0 \f(0)=(0)^3-2(0)^2-(0)+2=0-0-0+2=2 \f(1.5)=(1.5)^3-2(1.5)^2-(1.5)+2=-0.625 \f(2)=(2)^3-2(2)^2-(2)+2=8-8-2+2=0 \f(3)=(3)^3-2(3)^2-(3)+2=27-18-3+2=8$

Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.

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Question

Graph the function and identify its roots.

Answer

This question tests one's ability to graph a polynomial function.

For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior" falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).

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 algebraic technique to factor the function.

Recognize that the binomial is a perfect square for which the following formula can be used

$(a^2x^2-b^2)=(\sqrt{a}x-\sqrt{b})(\sqrt{a}x+\sqrt{b})$

since

$\a=4\Rightarrow \sqrt{a}=\pm2 \b=36\Rightarrow \sqrt{b}=\pm6$

thus the simplified, factored form is,

.

Step 2: Identify the roots of the function.

To find the roots of a function set its factored form equal to zero and solve for the possible x values.

$\2x-6=0 \2x=6 \x=3$ $\2x+6=0 \2x=-6 \x=-3$

Step 3: Create a table of pairs.

$\begin{tabular}{ |c|c|c|c|c|c|c| } \hline x & f(x) & y \ \hline -3& f(-3) & 0 \ -1 & f(-1) & -32 \ 0&f(0)& -36\ 1&f(1)&-32\3&f(3)&0 \\hline \end{tabular}$

The values in the table are found by substituting in the x values into the function as follows.

$\f(-3)=4(-3)^2-36=4(9)-36=36-36=0 \f(-1)=4(-1)^2-36=4-36=-32 \f(0)=4(0)^2-36=0-36=-36 \f(1)=4(1)^2-36=4-36=-32 \f(3)=4(3)^2-36=4(9)-36=36-36=0$

Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.

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Question

Graph the function and identify its roots.

Answer

This question tests one's ability to graph a polynomial function.

For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior" falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).

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 algebraic technique to factor the function.

Recognize that the binomial is a perfect square for which the following formula can be used

$(a^2x^2-b^2)=(\sqrt{a}x-\sqrt{b})(\sqrt{a}x+\sqrt{b})$

since

$\a=9\Rightarrow \sqrt{a}=\pm3 \b=16\Rightarrow \sqrt{b}=\pm4$

thus the simplified, factored form is,

.

Step 2: Identify the roots of the function.

To find the roots of a function set its factored form equal to zero and solve for the possible x values.

$\3x-4=0 \3x=4\\x=\frac{4}{3}$ $\3x+4=0 \3x=-4 \\x=-\frac{4}{3}$

Step 3: Create a table of pairs.

$\begin{tabular}{ |c|c|c|c|c|c|c| } \hline x & f(x) & y \ \hline -2& f(-2) & 20 \ -4/3 & f(-4/3) & 0 \ 0&f(0)& -16\ 1&f(1)&-7\\hline \end{tabular}$

The values in the table are found by substituting in the x values into the function as follows.

$\f(-2)=9(-2)^2-16=9(4)-16=36-16=20 \f(-4/3)=9(-4/3)^2-16=16-16=0 \f(0)=9(0)^2-10-16=-16 \f(1)=9(1)^2-16=9-16=-7$

Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.

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Question

Graph the function and identify the roots.

Answer

This question tests one's ability to graph a polynomial function.

For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior" falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).

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 algebraic technique to factor the function.

Recognize that the binomial is a perfect square for which the following formula can be used

$(a^2x^2-b^2)=(\sqrt{a}x-\sqrt{b})(\sqrt{a}x+\sqrt{b})$

since

$\a=100\Rightarrow \sqrt{a}=\pm10 \b=1\Rightarrow \sqrt{b}=\pm1$

thus the simplified, factored form is,

.

Step 2: Identify the roots of the function.

To find the roots of a function set its factored form equal to zero and solve for the possible x values.

$\10x-1=0 \10x=1\ \x=\frac{1}{10}$ $\10x+1=0 \10x=-1 \\x=-\frac{1}{10}$

Step 3: Create a table of pairs.

$\begin{tabular}{ |c|c|c|c|c|c|c| } \hline x & f(x) & y \ \hline -1& f(-1) & 99 \ 0 & f(0) & -1 \ 1&f(1)& 99 \\hline \end{tabular}$

The values in the table are found by substituting in the x values into the function as follows.

$\f(-1)=100(-1)^2-1=100-1=99 \f(0)=100(0)^2-1=0-1=-1 \f(1)=100(1)^2-1=100-1=99$

Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.

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Question

Write the quadratic function for the graph:

Answer

Method 1:

The x-intercepts are . These values would be obtained if the original quadratic were factored, or reverse-FOILed and the factors were set equal to zero.

For , . For , . These equations determine the resulting factors and the resulting function; .

Multiplying the factors and simplifying,

$f(x) = x\cdotx + 2\cdotx + 6\cdotx + 6\cdot2 = x^{2} + 8x + 12$ .

Answer: $f(x) = x^{2} + 8x + 12$ .

Method 2:

Use the form $(x - h)^{2} + k$ , where is the vertex.

is , so , .

$(x - [-4])^{2} + (-4) = (x + 4)^{2} - 4$

Answer: $f(x) = (x + 4)^{2} - 4$

$= x^{2} + 8x +16 - 4$

$= x^{2} + 8x + 12$

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Question

Write the quadratic function for the graph:

Answer

Because there are no x-intercepts, use the form $f(x) = (x - h)^{2} + k$ , where vertex is , so , , which gives

$f(x) = (x - 3)^{2} + 3$

$= x^{2} + 2(-3)(x) + (-3)^{2} + 3$

$= x^{2} - 6x + 9 + 3$

$= x^{2} - 6x + 12$

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Question

Which could be the equation for this graph?

Answer

This graph has zeros at 3, -2, and -4.5. This means that , , and . That last root is easier to work with if we consider it as $x+\frac{9}{2}=0$ and simplify it to . Also, this is a negative polynomial, because it is decreasing, increasing, decreasing and not the other way around.

Our equation results from multiplying , which results in .

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Question

Write the equation for the polynomial in this graph:

Answer

The zeros for this polynomial are $-1.5, \frac{1}{3}, 4$ .

This means that the factors are equal to zero when these values are plugged in for x.

multiply both sides by 2

so one factor is

$x - \frac{1}{3} = 0$ multiply both sides by 3

so one factor is

so one factor is

Multiply these three factors:

$(6x^2 + 7 x - 3 ) ( x - 4 ) = 6x^3 + 7x^2 - 3 x \enspace -24x^2 - 28x + 12$

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Question

Write the equation for the polynomial shown in this graph:

Answer

The zeros of this polynomial are . This means that the factors equal zero when these values are plugged in.

One factor is

One factor is

The third factor is equivalent to . Set equal to 0 and multiply by 2:

Multiply these three factors:

The graph is negative since it goes down then up then down, so we have to switch all of the signs:

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Question

Write the equation for the polynomial in this graph:

Answer

The zeros for this polynomial are $-1.75, \frac{1}{3}, 4$ . That means that the factors are equal to zero when these values are plugged in.

or equivalently $x + \frac{7}{4} = 0$ multiply both sides by 4

the first factor is

$x-\frac{1}{3} = 0$ multiply both sides by 3

the second factor is

the third factor is

Multiply the three factors:

$(12x^2 +17x - 7 )(x-4) = 12x^3 +17x^2 -7x + \enspace -48x^2 -68x +28$

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Question

Write the equation for the polynomial in the graph:

Answer

The zeros of the polynomial are . That means that the factors equal zero when these values are plugged in.

The first factor is or equivalently $x - \frac{3}{5} = 0$ multiply both sides by 5:

The second and third factors are and

Multiply:

$(5x^2 -18x + 9 ) ( x + 2 ) = 5x^3 -18x^2 +9x \enspace + 10x^2 -36x +18$

Because the graph goes down-up-down instead of the standard up-down-up, the graph is negative, so change all of the signs:

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Question

Which could be the equation for this graph?

Answer

This graph has zeros at 3, -2, and -4.5. This means that , , and . That last root is easier to work with if we consider it as $x+\frac{9}{2}=0$ and simplify it to . Also, this is a negative polynomial, because it is decreasing, increasing, decreasing and not the other way around.

Our equation results from multiplying , which results in .

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Question

Write the quadratic function for the graph:

Answer

Method 1:

The x-intercepts are . These values would be obtained if the original quadratic were factored, or reverse-FOILed and the factors were set equal to zero.

For , . For , . These equations determine the resulting factors and the resulting function; .

Multiplying the factors and simplifying,

$f(x) = x\cdotx + 2\cdotx + 6\cdotx + 6\cdot2 = x^{2} + 8x + 12$ .

Answer: $f(x) = x^{2} + 8x + 12$ .

Method 2:

Use the form $(x - h)^{2} + k$ , where is the vertex.

is , so , .

$(x - [-4])^{2} + (-4) = (x + 4)^{2} - 4$

Answer: $f(x) = (x + 4)^{2} - 4$

$= x^{2} + 8x +16 - 4$

$= x^{2} + 8x + 12$

Compare your answer with the correct one above

Question

Write the quadratic function for the graph:

Answer

Because there are no x-intercepts, use the form $f(x) = (x - h)^{2} + k$ , where vertex is , so , , which gives

$f(x) = (x - 3)^{2} + 3$

$= x^{2} + 2(-3)(x) + (-3)^{2} + 3$

$= x^{2} - 6x + 9 + 3$

$= x^{2} - 6x + 12$

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Question

Write the equation for the polynomial in this graph:

Answer

The zeros for this polynomial are $-1.5, \frac{1}{3}, 4$ .

This means that the factors are equal to zero when these values are plugged in for x.

multiply both sides by 2

so one factor is

$x - \frac{1}{3} = 0$ multiply both sides by 3

so one factor is

so one factor is

Multiply these three factors:

$(6x^2 + 7 x - 3 ) ( x - 4 ) = 6x^3 + 7x^2 - 3 x \enspace -24x^2 - 28x + 12$

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Question

Write the equation for the polynomial shown in this graph:

Answer

The zeros of this polynomial are . This means that the factors equal zero when these values are plugged in.

One factor is

One factor is

The third factor is equivalent to . Set equal to 0 and multiply by 2:

Multiply these three factors:

The graph is negative since it goes down then up then down, so we have to switch all of the signs:

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Question

Write the equation for the polynomial in this graph:

Answer

The zeros for this polynomial are $-1.75, \frac{1}{3}, 4$ . That means that the factors are equal to zero when these values are plugged in.

or equivalently $x + \frac{7}{4} = 0$ multiply both sides by 4

the first factor is

$x-\frac{1}{3} = 0$ multiply both sides by 3

the second factor is

the third factor is

Multiply the three factors:

$(12x^2 +17x - 7 )(x-4) = 12x^3 +17x^2 -7x + \enspace -48x^2 -68x +28$

Compare your answer with the correct one above

Question

Write the equation for the polynomial in the graph:

Answer

The zeros of the polynomial are . That means that the factors equal zero when these values are plugged in.

The first factor is or equivalently $x - \frac{3}{5} = 0$ multiply both sides by 5:

The second and third factors are and

Multiply:

$(5x^2 -18x + 9 ) ( x + 2 ) = 5x^3 -18x^2 +9x \enspace + 10x^2 -36x +18$

Because the graph goes down-up-down instead of the standard up-down-up, the graph is negative, so change all of the signs:

Compare your answer with the correct one above

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