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All New SAT Math - No Calculator Resources

3 Diagnostic Tests 1 Practice Test Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #52 : Seeing Structure In Expressions

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

x^2-6x-6

Possible Answers:

(-3,15)

(-15,3)

(3,-15)

(-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,

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

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.

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

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 #7 : 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^2-2x-1

Possible Answers:

(0,1)

(-1,0)

(1,0)

(0,-1)

(0,0)

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 #8 : 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:

(3,-2)

(2,-3)

(2,3)

(-2,-3)

(-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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Example Question #9 : 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-1

Possible Answers:

(-3,8)

(3,8)

(-3,-8)

(8,-3)

(3,-8)

Correct answer:

(-3,8)

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-1

First factor out a negative one.

-1(x^2+6x+1)

Now 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.

$-(x^2+6x+1)\Rightarrow-((x+3)^2+1+3^2)$

When simplified the new function is,

$\\-((x+3)^2+10)$

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

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

-x^2-6x-1

$\\y=-(-3)^2-6(-3)-1 \\y=-9+18-1 \\y=8$

Therefore the maximum value occurs at the point (-3,8) .

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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.

f(x)=-x^2-8x-2

Possible Answers:

(-4,14)

(4,-14)

(-4,-14)

(14,-4)

(4,14)

Correct answer:

(-4,14)

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-2

First factor out a negative one.

-1(x^2+8x+2)

Now 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.

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

When simplified the new function is,

$\\-((x+4)^2+18)$

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

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

-x^2-8x-2

$\\y=-(-4)^2-8(-4)-2 \\y=-16+32-2 \\y=14$

Therefore the maximum value occurs at the point (-4,14) .

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Example Question #10 : 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-5

Possible Answers:

(2,-1)

(-2,-1)

(2,1)

(-2,1)

(-1,-2)

Correct answer:

(-2,-1)

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-5

First factor out a negative one.

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

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.

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

When simplified the new function is,

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

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

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

-x^2-4x-5

$\\y=-(-2)^2-4(-2)-5 \\y=-4+8-5 \\y=-1$

Therefore the maximum value occurs at the point (-2,-1) .

Report an Error

Example Question #22 : Graphical Representation Of Functions

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

-x^2-6x-5

Possible Answers:

(-3,-4)

(4,-3)

(-3,4)

(3,4)

(3,-4)

Correct answer:

(-3,4)

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-5

First factor out a negative one.

-1(x^2+6x+5)

Now 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.

$-(x^2+6x+5)\Rightarrow-((x+3)^2+5+3^2)$

When simplified the new function is,

$\\-((x+3)^2+14)$

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

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

-x^2-6x-1

$\\y=-(-3)^2-6(-3)-5 \\y=-9+18-5 \\y=4$

Therefore the maximum value occurs at the point (-3,4) .

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Example Question #551 : New Sat

What does the scale factor of a dilation need to be to ensure that triangles are not only similar but also congruent?

Possible Answers:

$\frac{1}{2}$

Correct answer:

Explanation:

The scale factor of a dilation tells us what we multiply corresponding sides by to get the new side lengths. In this case, we want these lengths to be the same to get congruent triangles. Thus, we must be looking for the multiplicative identity, which is 1.

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Example Question #1 : Triangles

Which of the following is not a theorem to prove that triangles are similar?

Possible Answers:

ASA

SSS

SAS

Correct answer:

ASA

Explanation:

ASA (Angle Side Angle) is a theorem to prove triangle congruency.

In this case, we only need two angles to prove that two triangles are similar, so the last side in ASA is unnecessary for this question.

For this purpose, we use the theorem AA instead.

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Example Question #2 : Triangles

One triangle has side measures , , and . Another has side lengths , , and . Are these triangles similar?

Possible Answers:

Those can't be the side lengths of triangles.

There is not enough information given.

Triangles can't be similar!

Yes

Correct answer:

Explanation:

Two triangles are similar if and only if their side lengths are proportional.

In this case, two of the sides are proportional, leading us to a scale factor of 2.

However, with the last side, $8\cdot 2 = 16$ which is not our side length.

Thus, these pair of sides are not proportional and therefore our triangles cannot be similar.

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New SAT Math - No Calculator : New SAT

Study concepts, example questions & explanations for New SAT Math - No Calculator

All New SAT Math - No Calculator Resources

Example Questions

Example Question #52 : Seeing Structure In Expressions

Example Question #7 : Calculate Maximum Or Minimum Of Quadratic By Completing The Square: Ccss.Math.Content.Hsa Sse.B.3b

Example Question #8 : Calculate Maximum Or Minimum Of Quadratic By Completing The Square: Ccss.Math.Content.Hsa Sse.B.3b

Example Question #9 : 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 #10 : Calculate Maximum Or Minimum Of Quadratic By Completing The Square: Ccss.Math.Content.Hsa Sse.B.3b

Example Question #22 : Graphical Representation Of Functions

Example Question #551 : New Sat

Example Question #1 : Triangles

Example Question #2 : Triangles

All New SAT Math - No Calculator Resources

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