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Algebra 1 : Graphing

Study concepts, example questions & explanations for Algebra 1

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Example Questions

Example Question #1 : How To Graph A Quadratic Function

What is the equation of a parabola with vertex (4,1) and -intercept (0,-7) ?

Possible Answers:

$y = \frac{1}{2} x^{2} -4x - 7$

$y = -\frac{1}{2} x^{2} +4x - 7$

$y = \frac{1}{2} x^{2} +8x - 7$

$y = \frac{1}{2} x^{2} -8x - 7$

$y = -\frac{1}{2} x^{2} -4x - 7$

Correct answer:

$y = -\frac{1}{2} x^{2} +4x - 7$

Explanation:

From the vertex, we know that the equation of the parabola will take the form $y = a (x - 4) ^{2}+ 1$ for some .

To calculate that , we plug in the values from the other point we are given, x = 0, y=-7 , and solve for :

$y = a (x - 4)^{2} + 1$

$-7= a \cdot (0 - 4)^{2} + 1$

$-7= a \cdot (- 4)^{2} + 1$

-7= 16a + 1

16a = -8

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

Now the equation is $y = -\frac{1}{2} (x - 4)^{2} + 1$ . This is not an answer choice, so we need to rewrite it in some way.

Expand the squared term:

$y = -\frac{1}{2} (x^{2} -8x + 16) + 1$

Distribute the fraction through the parentheses:

$y = -\frac{1}{2} x^{2} +4x - 8 + 1$

Combine like terms:

$y = -\frac{1}{2} x^{2} +4x - 7$

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Example Question #1 : How To Graph A Quadratic Function

Possible Answers:

$y = x^{2}$

$y = (x-2)^{2}$

$y = -(x-2)^{2} - 2$

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

None of the above

Correct answer:

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

Explanation:

Starting with $y = x^{2}$

$(x-2)^{2}$ moves the parabola $x^{2}$ by units to the right.

Similarly $(x+3)^{2}$ moves the parabola by units to the left.

Hence the correct answer is option .

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Example Question #41 : Graphing

Which of the following graphs matches the function y=(x-1)^2-2 ?

Possible Answers:

Graph4

Graph3

Graph1

Graph

Graph2

Correct answer:

Graph

Explanation:

Start by visualizing the graph associated with the function :

Graph5

Terms within the parentheses associated with the squared x-variable will shift the parabola horizontally, while terms outside of the parentheses will shift the parabola vertically. In the provided equation, 2 is located outside of the parentheses and is subtracted from the terms located within the parentheses; therefore, the parabola in the graph will shift down by 2 units. A simplified graph of looks like this:

Graph6

Remember that there is also a term within the parentheses. Within the parentheses, 1 is subtracted from the x-variable; thus, the parabola in the graph will shift to the right by 1 unit. As a result, the following graph matches the given function :

Graph

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Example Question #1 : How To Graph An Absolute Value Function

Absolute_value

Which of these would most likely be the equation corresponding to the above graph?

Possible Answers:

y = |x + 2 | +1

y = |x + 3 |

y = |x + 2 | -1

y = |x - 2 | +1

y = |x - 2 | -1

Correct answer:

y = |x - 2 | -1

Explanation:

This is an absolute value graph. Its equation takes the form y = |x-h|+k , in which h,k represent the number of units that the base graph y = |x| is translated right and up respectively.

Since the graph of y = |x| is translated two units right and one unit down, h = 2 and k = -1 , so the equation would be:

y = |x-2|+(-1)

y = |x - 2 | -1

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Example Question #2 : How To Graph An Absolute Value Function

Give the -intercept(s) of the graph of the function $f(x) = 7 - \left | 2x-3 \right |$

Possible Answers:

(-2,0) , (5,0)

The graph has no -intercepts.

(-2,0) , (2,0)

(-5,0) , (2,0)

(-5,0) , (5,0)

Correct answer:

(-2,0) , (5,0)

Explanation:

To find the -intercept(s) of the graph, set f(x) = 0 and solve for .

$f(x) = 7 - \left | 2x-3 \right |$

$0= 7 - \left | 2x-3 \right |$

$\left | 2x-3 \right |= 7$

Rewrite this as the compound equation:

2x-3= -7 or 2x-3= 7

Solve each separately:

2x-3= -7

2x-3 + 3 = -7+ 3

2x = -4

$2x \div 2 = -4\div 2$

x = -2

2x-3= 7

2x-3 + 3 = 7+ 3

2x = 10

$2x \div 2 = 10\div 2$

x = 5

There are two -intercepts: (-2,0) , (5,0)

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Example Question #1 : How To Graph An Absolute Value Function

Absolute_value

Which of these would most likely be the equation corresponding to the above graph?

Possible Answers:

y = | x - 6 | - 3

y = | x - 3 | - 6

y = | x - 6 | + 3

y = | x + 3 | - 6

y = | x + 6 | - 3

Correct answer:

y = | x + 3 | - 6

Explanation:

This is an absolute value graph. Its equation takes the form y = |x-h|+k , in which h,k represent the number of units that the base graph y = |x| is translated right and up respectively.

Since the graph of y = |x| is translated three units left and six units down, h = -3 and k = -6 .

Plug these values into the general form of the equation:

y = |x-(-3)|+(-6)

Simplify:

y = |x+3|-6

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Example Question #1 : How To Graph An Absolute Value Function

$\textup{What is the slope of } y=-4\left | x-2\right | \textup{ when } x=1?$

Possible Answers:

$\textup{It does not exist.}$

Correct answer:

Explanation:

$\textup{When }x=1\textup{ the expression inside the absolute value is negative,} \\\textup{so the function can be written as } y=-4(2-x)=4x-8,\textup{ with a slope of 4.}\\\textup{Plus, the slope changes to -4 when } x>2. \textup{ When } x=2 \textup{ it does not exist.}$

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Example Question #2 : How To Graph An Absolute Value Function

$\textup{Find the lowest attainable value of }y \textup{ in } y=-4+ \left |4x^2-25 \right |.$

Possible Answers:

2.5

-2.5

Correct answer:

Explanation:

$\\ \left | 4x^2-25\right | \textup{cannot get any lower than zero,}\\\textup{so\textit{ y} cannot get any lower than} -4.$

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Example Question #2 : How To Graph An Absolute Value Function

$\textup{Which is an \textit{x}-intercept of the function } y=\left | -x^2-8x-16\right |-4.$

Possible Answers:

$\textup{None of the above.}$

Correct answer:

Explanation:

$\\ \textup{The function can be simplified as } \\y=\left | -(x+4)^2\right |-4=\left |(x+4)^2\right |-4=(x+4)^2-4.\\\textup{In order to find the \textit{x}-intercecpt, we solve the equation}\\0=(x+4)^2-4 \textup{ which becomes }\\(x+4)=\pm2\textup{ and has solution } -2 \textup{ and } -6.$

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Algebra 1 : Graphing

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #1 : How To Graph A Quadratic Function

Example Question #1 : How To Graph A Quadratic Function

Example Question #41 : Graphing

Example Question #1 : How To Graph An Absolute Value Function

Example Question #2 : How To Graph An Absolute Value Function

Example Question #1 : How To Graph An Absolute Value Function

Example Question #1 : How To Graph An Absolute Value Function

Example Question #2 : How To Graph An Absolute Value Function

Example Question #2 : How To Graph An Absolute Value Function

All Algebra 1 Resources

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