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New SAT Math - Calculator : Graphical Representation of Functions

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Example Questions

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Example Question #11 : Finding Zeros

Find all possible zeros for the following function.

x^2-x

Possible Answers:

x=-1

x=0,1

x=1

x=0

x=-1,0

Correct answer:

x=0,1

Explanation:

To find the zeros of this function first identify and factor of the GCF.

x^2-x

In this particular case,the GCF is as it appears in both terms. Factoring out the GCF results in the following.

$\\x^2-x \\x\cdot x-x\cdot 1 \\x(x-1)$

From here, set each term equal to zero and solve for .

x=0

and

$\\x-1=0 \\x-1+1=0+1 \\x=1$

Q12

Therefore the zeros are,

x=0,1

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Example Question #12 : Finding Zeros

Find all possible zeros for the following function.

x^2+8x+16

Possible Answers:

x=1

x=0

x=-4,4

x=4

x=-4

Correct answer:

x=-4

Explanation:

To find the zeros of the function use factoring.

x^2+8x+16

Set up the expression in factored form, leaving blanks for the numbers that are not yet known.

$(x+\underline{\hspace{.5cm}})(x+\underline{\hspace{.5cm}})$

At this point, you need to find two numbers - one for each blank. By looking at the original expression, a few clues can be gathered that will help find the two numbers. The product of these two numbers will be equal to the last term of the original expression (16, or c in the standard quadratic formula), and their sum will be equal to the coefficient of the second term of the original expression (8, or b in the standard quadratic formula). Because their product is positive (16) and the sum is positive, that must mean that they both have positive signs.

Now, at this point, test a few different possibilities using the clues gathered from the original expression. In the end, it's found that the only numbers that work are 4 and 4, as the product of 4 and 4 is 16, and sum of 4 and 4 is 8. So, this results in the expression's factored form looking like...

(x+4)(x+4)

From here, set the binomial equal to zero and solve for .

$\\x+4=0 \\x+4-4=-4 \\x=-4$

To verify the zeros, graph the original function and identify where the graph touches or crosses the x-axis.

Q11

Therefore the zero of the function is,

x=-4

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Example Question #191 : Equations / Inequalities

Complete the square to calculate the maximum or minimum point of the given function.

x^2+4x+7

Possible Answers:

$\text{Maximum}=(-2,-3)$

$\text{Maximum}=(-2,3)$

$\text{Minimum}=(-2,-3)$

$\text{Minimum}=(2,3)$

$\text{Minimum}=(-2,3)$

Correct answer:

$\text{Minimum}=(-2,3)$

Explanation:

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

a^2x^2+2abx+b^2

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

x^2+4x+7

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 4$

Now divide the middle term coefficient by two.

$\frac{4}{2}=2$

From here write the function with the perfect square. Remember when adding the new squared term, add it to both sides to keep the equation balanced.

$x^2+4x+7=0\Rightarrow (x+2)^2+2^2+7=0+2^2$

When simplified the new function is,

$\\(x+2)^2+4+7=4 \\(x+2)^2+11=4 \\(x+2)^2=-7$

Since the x^2 term is positive, the parabola will be opening up. This means that the function has a minimum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

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

From here, substitute the the value into the original function.

x^2+4x+7

$\\y=(-2)^2+4(-2)+7 \\y=4-8+7 \\y=3$

Therefore the minimum value occurs at the point (-2,3) .

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Example Question #32 : Quadratic Equations

Complete the square to calculate the maximum or minimum point of the given function.

x^2+2x-4

Possible Answers:

(-1,5)

(1,-5)

(-5,-1)

(-1,-5)

(1,5)

Correct answer:

(-1,-5)

Explanation:

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

a^2x^2+2abx+b^2

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

x^2+2x-4

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 2$

Now divide the middle term coefficient by two.

$\frac{2}{2}=1$

From here write the function with the perfect square. Remember when adding the new squared term, add it to both sides to keep the equation balanced.

$x^2+2x-4=0\Rightarrow (x+1)^2+1^2-4=0+1^2$

When simplified the new function is,

$\\(x+1)^2-4+1 \\(x+1)^2-3$

$\\x+1=0 \\x+1-1=0-1 \\x=-1$

From here, substitute the the value into the original function.

x^2+2x-4

$\\y=(-1)^2+2(-1)-4 \\y=1-2-4 \\y=-5$

Therefore the minimum value occurs at the point (-1,-5) .

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Example Question #1 : Calculate Maximum Or Minimum Of Quadratic By Completing The Square: Ccss.Math.Content.Hsa Sse.B.3b

Complete the square to calculate the maximum or minimum point of the given function.

x^2+8x-3

Possible Answers:

(-4,-19)

(4,-19)

(-19,-4)

(4,19)

(-4,19)

Correct answer:

(-4,-19)

Explanation:

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

a^2x^2+2abx+b^2

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

x^2+8x-3

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 8$

Now divide the middle term coefficient by two.

$\frac{8}{2}=4$

From here write the function with the perfect square.

$x^2+8x-3\Rightarrow (x+4)^2-3+4^2$

When simplified the new function is,

$\\(x+4)^2-3+16\\(x+4)^2+13$

$\\x+4=0 \\x+4-4=0-4 \\x=-4$

From here, substitute the the value into the original function.

x^2+8x-3

$\\y=(-4)^2+8(-4)-3 \\y=16-32-3 \\y=-19$

Therefore the minimum value occurs at the point (-4,-19) .

Report an Error

Example Question #51 : Seeing Structure In Expressions

Complete the square to calculate the maximum or minimum point of the given function.

x^2+6x+2

Possible Answers:

(-3,7)

(-7,-3)

(3,7)

(3,-7)

(-3,-7)

Correct answer:

(-3,-7)

Explanation:

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

a^2x^2+2abx+b^2

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

x^2+6x+2

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= 6$

Now divide the middle term coefficient by two.

$\frac{6}{2}=3$

From here write the function with the perfect square.

$x^2+6x+2\Rightarrow (x+3)^2+6+3^2$

When simplified the new function is,

$\\(x+3)^2+2+9\\(x+3)^2+11$

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

From here, substitute the the value into the original function.

x^2+6x+2

$\\y=(-3)^2+6(-3)+2 \\y=9-18+2 \\y=-7$

Therefore the minimum value occurs at the point (-3,-7) .

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Example Question #2 : Calculate Maximum Or Minimum Of Quadratic By Completing The Square: Ccss.Math.Content.Hsa Sse.B.3b

Complete the square to calculate the maximum or minimum point of the given function.

x^2-4x+1

Possible Answers:

(-2,3)

(-2,-3)

(2,3)

(2,-3)

(-3,2)

Correct answer:

(2,-3)

Explanation:

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

a^2x^2+2abx+b^2

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

x^2-4x+1

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= -4$

Now divide the middle term coefficient by two.

$\frac{-4}{2}=-2$

From here write the function with the perfect square.

$x^2-4x+1\Rightarrow (x-2)^2+1+(-2)^2$

When simplified the new function is,

$\\(x-2)^2+1+4\\(x-2)^2+5$

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

From here, substitute the the value into the original function.

x^2-4x+1

$\\y=(2)^2-4(2)+1 \\y=4-8+1 \\y=-3$

Therefore the minimum value occurs at the point (2,-3) .

Report an Error

Example Question #6 : Calculate Maximum Or Minimum Of Quadratic By Completing The Square: Ccss.Math.Content.Hsa Sse.B.3b

Complete the square to calculate the maximum or minimum point of the given function.

x^2-6x-6

Possible Answers:

(-3,-15)

(-3,15)

(-15,3)

(3,15)

(3,-15)

Correct answer:

(3,-15)

Explanation:

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

a^2x^2+2abx+b^2

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

x^2-6x-6

First identify the middle term coefficient.

$\text{Middle Term Coefficient}= -6$

Now divide the middle term coefficient by two.

$\frac{-6}{2}=-3$

From here write the function with the perfect square.

$x^2-6x-6\Rightarrow (x-3)^2-6+(-3)^2$

When simplified the new function is,

$\\(x-3)^2-6+9\\(x-3)^2+3$

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

From here, substitute the the value into the original function.

x^2-6x-6

$\\y=(3)^2-6(3)-6 \\y=9-18-6 \\y=-15$

Therefore the minimum value occurs at the point (3,-15) .

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Example Question #51 : Seeing Structure In Expressions

Complete the square to calculate the maximum or minimum point of the given function.

-x^2-2x-1

Possible Answers:

(-1,0)

(0,0)

(0,-1)

(1,0)

(0,1)

Correct answer:

(-1,0)

Explanation:

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

a^2x^2+2abx+b^2

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

-x^2-2x-1

First factor out a negative one.

-1(x^2+2x+1)

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 2$

Now divide the middle term coefficient by two.

$\frac{2}{2}=1$

From here write the function with the perfect square.

$-(x^2+2x+1)\Rightarrow-((x+1)^2+1+1)$

When simplified the new function is,

$\\-((x+1)^2+2)$

Since the x^2 term is negative, the parabola will be opening down. This means that the function has a maximum value at the vertex. To find the value of the vertex set the inside portion of the binomial equal to zero and solve.

$\\x+1=0 \\x+1-1=0-1 \\x=-1$

From here, substitute the the value into the original function.

-x^2-2x-1

$\\y=-(-1)^2-2(-1)-1 \\y=-1+2-1 \\y=0$

Therefore the maximum value occurs at the point (-1,0) .

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Example Question #1 : Graphing Quadratics & Polynomials

Complete the square to calculate the maximum or minimum point of the given function.

-x^2-4x-1

Possible Answers:

(-2,-3)

(2,3)

(2,-3)

(3,-2)

(-2,3)

Correct answer:

(-2,3)

Explanation:

Completing the square method uses the concept of perfect squares. Recall that a perfect square is in the form,

$(ax\pm b)^2=(ax\pm b)(ax\pm b)$

where when multiplied out,

a^2x^2+2abx+b^2

the middle term coefficient, when divided by two and squared, results in the coefficient of the last term.

$2ab=\left(\frac{2ab}{2} \right )^2=b^2$

Complete the square for this particular function is as follows.

-x^2-4x-1

First factor out a negative one.

-1(x^2+4x+1)

Now identify the middle term coefficient.

$\text{Middle Term Coefficient}= 4$

Now divide the middle term coefficient by two.

$\frac{4}{2}=2$

From here write the function with the perfect square.

$-(x^2+4x+1)\Rightarrow-((x+2)^2+1+2^2)$

When simplified the new function is,

$\\-((x+2)^2+5)$

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

From here, substitute the the value into the original function.

-x^2-4x-1

$\\y=-(-2)^2-4(-2)-1 \\y=-4+8-1 \\y=3$

Therefore the maximum value occurs at the point (-2,3) .

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New SAT Math - Calculator : Graphical Representation of Functions

Study concepts, example questions & explanations for New SAT Math - Calculator

All New SAT Math - Calculator Resources

Example Questions

Example Question #11 : Finding Zeros

Example Question #12 : Finding Zeros

Example Question #191 : Equations / Inequalities

Example Question #32 : Quadratic Equations

Example Question #1 : Calculate Maximum Or Minimum Of Quadratic By Completing The Square: Ccss.Math.Content.Hsa Sse.B.3b

Example Question #51 : Seeing Structure In Expressions

Example Question #2 : Calculate Maximum Or Minimum Of Quadratic By Completing The Square: Ccss.Math.Content.Hsa Sse.B.3b

Example Question #6 : Calculate Maximum Or Minimum Of Quadratic By Completing The Square: Ccss.Math.Content.Hsa Sse.B.3b

Example Question #51 : Seeing Structure In Expressions

Example Question #1 : Graphing Quadratics & Polynomials

All New SAT Math - Calculator Resources

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