Common Core: High School - Algebra : Seeing Structure in Expressions

Study concepts, example questions & explanations for Common Core: High School - Algebra

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All Common Core: High School - Algebra Resources

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Example Questions

Example Question #151 : New Sat Math Calculator

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

\displaystyle x^2+8x-3

Possible Answers:

\displaystyle (4,19)

\displaystyle (-4,-19)

\displaystyle (-19,-4)

\displaystyle (4,-19)

\displaystyle (-4,19)

Correct answer:

\displaystyle (-4,-19)

Explanation:

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

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

where when multiplied out,

\displaystyle a^2x^2+2abx+b^2

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

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

Complete the square for this particular function is as follows.

\displaystyle x^2+8x-3

First identify the middle term coefficient.

\displaystyle \text{Middle Term Coefficient}= 8

Now divide the middle term coefficient by two.

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

From here write the function with the perfect square.

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

When simplified the new function is,

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

Since the \displaystyle 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 \displaystyle x value of the vertex set the inside portion of the binomial equal to zero and solve.

\displaystyle \\x+4=0 \\x+4-4=0-4 \\x=-4

From here, substitute the the \displaystyle x value into the original function.

\displaystyle x^2+8x-3

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

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

Example Question #152 : New Sat Math Calculator

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

\displaystyle x^2+6x+2

Possible Answers:

\displaystyle (3,7)

\displaystyle (3,-7)

\displaystyle (-7,-3)

\displaystyle (-3,7)

\displaystyle (-3,-7)

Correct answer:

\displaystyle (-3,-7)

Explanation:

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

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

where when multiplied out,

\displaystyle a^2x^2+2abx+b^2

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

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

Complete the square for this particular function is as follows.

\displaystyle x^2+6x+2

First identify the middle term coefficient.

\displaystyle \text{Middle Term Coefficient}= 6

Now divide the middle term coefficient by two.

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

From here write the function with the perfect square.

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

When simplified the new function is,

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

Since the \displaystyle 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 \displaystyle x value of the vertex set the inside portion of the binomial equal to zero and solve.

\displaystyle \\x+3=0 \\x+3-3=0-3 \\x=-3

From here, substitute the the \displaystyle x value into the original function.

\displaystyle x^2+6x+2

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

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

Example Question #2 : Graphing Quadratics & Polynomials

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

\displaystyle x^2-4x+1

Possible Answers:

\displaystyle (-3,2)

\displaystyle (2,3)

\displaystyle (-2,3)

\displaystyle (2,-3)

\displaystyle (-2,-3)

Correct answer:

\displaystyle (2,-3)

Explanation:

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

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

where when multiplied out,

\displaystyle a^2x^2+2abx+b^2

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

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

Complete the square for this particular function is as follows.

\displaystyle x^2-4x+1

First identify the middle term coefficient.

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

Now divide the middle term coefficient by two.

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

From here write the function with the perfect square.

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

When simplified the new function is,

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

Since the \displaystyle 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 \displaystyle x value of the vertex set the inside portion of the binomial equal to zero and solve.

\displaystyle \\x-2=0 \\x-2+2=0+2\\x=2

From here, substitute the the \displaystyle x value into the original function.

\displaystyle x^2-4x+1

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

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

Example Question #3 : Graphing Quadratics & Polynomials

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

\displaystyle x^2-6x-6

Possible Answers:

\displaystyle (-15,3)

\displaystyle (-3,-15)

\displaystyle (3,-15)

\displaystyle (-3,15)

\displaystyle (3,15)

Correct answer:

\displaystyle (3,-15)

Explanation:

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

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

where when multiplied out,

\displaystyle a^2x^2+2abx+b^2

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

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

Complete the square for this particular function is as follows.

\displaystyle x^2-6x-6

First identify the middle term coefficient.

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

Now divide the middle term coefficient by two.

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

From here write the function with the perfect square.

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

When simplified the new function is,

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

Since the \displaystyle 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 \displaystyle x value of the vertex set the inside portion of the binomial equal to zero and solve.

\displaystyle \\x-3=0 \\x-3+3=0+3\\x=3

From here, substitute the the \displaystyle x value into the original function.

\displaystyle x^2-6x-6

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

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

Example Question #4 : Graphing Quadratics & Polynomials

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

\displaystyle -x^2-2x-1

Possible Answers:

\displaystyle (0,0)

\displaystyle (1,0)

\displaystyle (0,-1)

\displaystyle (-1,0)

\displaystyle (0,1)

Correct answer:

\displaystyle (-1,0)

Explanation:

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

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

where when multiplied out,

\displaystyle a^2x^2+2abx+b^2

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

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

Complete the square for this particular function is as follows.

\displaystyle -x^2-2x-1

First factor out a negative one.

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

Now identify the middle term coefficient.

\displaystyle \text{Middle Term Coefficient}= 2

Now divide the middle term coefficient by two.

\displaystyle \frac{2}{2}=1

From here write the function with the perfect square.

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

When simplified the new function is,

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

Since the \displaystyle 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 \displaystyle x value of the vertex set the inside portion of the binomial equal to zero and solve.

\displaystyle \\x+1=0 \\x+1-1=0-1 \\x=-1

From here, substitute the the \displaystyle x value into the original function.

\displaystyle -x^2-2x-1

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

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

Example Question #151 : New Sat Math Calculator

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

\displaystyle -x^2-4x-1

Possible Answers:

\displaystyle (3,-2)

\displaystyle (-2,-3)

\displaystyle (2,3)

\displaystyle (2,-3)

\displaystyle (-2,3)

Correct answer:

\displaystyle (-2,3)

Explanation:

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

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

where when multiplied out,

\displaystyle a^2x^2+2abx+b^2

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

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

Complete the square for this particular function is as follows.

\displaystyle -x^2-4x-1

First factor out a negative one.

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

Now identify the middle term coefficient.

\displaystyle \text{Middle Term Coefficient}= 4

Now divide the middle term coefficient by two.

\displaystyle \frac{4}{2}=2

From here write the function with the perfect square.

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

When simplified the new function is,

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

Since the \displaystyle 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 \displaystyle x value of the vertex set the inside portion of the binomial equal to zero and solve.

\displaystyle \\x+2=0 \\x+2-2=0-2 \\x=-2

From here, substitute the the \displaystyle x value into the original function.

\displaystyle -x^2-4x-1

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

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

Example Question #5 : Graphing Quadratics & Polynomials

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

\displaystyle -x^2-6x-1

Possible Answers:

\displaystyle (3,8)

\displaystyle (-3,-8)

\displaystyle (-3,8)

\displaystyle (8,-3)

\displaystyle (3,-8)

Correct answer:

\displaystyle (-3,8)

Explanation:

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

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

where when multiplied out,

\displaystyle a^2x^2+2abx+b^2

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

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

Complete the square for this particular function is as follows.

\displaystyle -x^2-6x-1

First factor out a negative one.

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

Now identify the middle term coefficient.

\displaystyle \text{Middle Term Coefficient}= 6

Now divide the middle term coefficient by two.

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

From here write the function with the perfect square.

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

When simplified the new function is,

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

Since the \displaystyle 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 \displaystyle x value of the vertex set the inside portion of the binomial equal to zero and solve.

\displaystyle \\x+3=0 \\x+3-3=0-3 \\x=-3

From here, substitute the the \displaystyle x value into the original function.

\displaystyle -x^2-6x-1

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

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

Example Question #161 : New Sat Math Calculator

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

\displaystyle -x^2-4x-5

Possible Answers:

\displaystyle (-2,-1)

\displaystyle (2,-1)

\displaystyle (-1,-2)

\displaystyle (-2,1)

\displaystyle (2,1)

Correct answer:

\displaystyle (-2,-1)

Explanation:

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

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

where when multiplied out,

\displaystyle a^2x^2+2abx+b^2

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

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

Complete the square for this particular function is as follows.

\displaystyle -x^2-4x-5

First factor out a negative one.

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

Now identify the middle term coefficient.

\displaystyle \text{Middle Term Coefficient}= 4

Now divide the middle term coefficient by two.

\displaystyle \frac{4}{2}=2

From here write the function with the perfect square.

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

When simplified the new function is,

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

Since the \displaystyle 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 \displaystyle x value of the vertex set the inside portion of the binomial equal to zero and solve.

\displaystyle \\x+2=0 \\x+2-2=0-2 \\x=-2

From here, substitute the the \displaystyle x value into the original function.

\displaystyle -x^2-4x-5

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

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

Example Question #41 : Quadratic Equations

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

\displaystyle -x^2-6x-5

Possible Answers:

\displaystyle (3,-4)

\displaystyle (-3,-4)

\displaystyle (3,4)

\displaystyle (4,-3)

\displaystyle (-3,4)

Correct answer:

\displaystyle (-3,4)

Explanation:

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

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

where when multiplied out,

\displaystyle a^2x^2+2abx+b^2

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

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

Complete the square for this particular function is as follows.

\displaystyle -x^2-6x-5

First factor out a negative one.

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

Now identify the middle term coefficient.

\displaystyle \text{Middle Term Coefficient}= 6

Now divide the middle term coefficient by two.

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

From here write the function with the perfect square.

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

When simplified the new function is,

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

Since the \displaystyle 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 \displaystyle x value of the vertex set the inside portion of the binomial equal to zero and solve.

\displaystyle \\x+3=0 \\x+3-3=0-3 \\x=-3

From here, substitute the the \displaystyle x value into the original function.

\displaystyle -x^2-6x-1

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

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

Example Question #1 : Graphing Quadratics & Polynomials

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

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

Possible Answers:

\displaystyle (-4,-14)

\displaystyle (4,14)

\displaystyle (-4,14)

\displaystyle (4,-14)

\displaystyle (14,-4)

Correct answer:

\displaystyle (-4,14)

Explanation:

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

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

where when multiplied out,

\displaystyle a^2x^2+2abx+b^2

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

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

Complete the square for this particular function is as follows.

\displaystyle -x^2-8x-2

First factor out a negative one.

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

Now identify the middle term coefficient.

\displaystyle \text{Middle Term Coefficient}= 8

Now divide the middle term coefficient by two.

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

From here write the function with the perfect square.

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

When simplified the new function is,

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

Since the \displaystyle 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 \displaystyle x value of the vertex set the inside portion of the binomial equal to zero and solve.

\displaystyle \\x+4=0 \\x+4-4=0-4 \\x=-4

From here, substitute the the \displaystyle x value into the original function.

\displaystyle -x^2-8x-2

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

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

All Common Core: High School - Algebra Resources

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