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Algebra 1 : How to graph a quadratic function

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Example Questions

Example Question #1 : Quadratic Functions

What is the minimum possible value of the expression below?

$3x^{2}+6x-10$

Possible Answers:

The expression has no minimum value.

Correct answer:

Explanation:

We can determine the lowest possible value of the expression by finding the -coordinate of the vertex of the parabola graphed from the equation $y=3x^{2}+6x-10$ . This is done by rewriting the equation in vertex form.

$3x^{2}+6x-10$

$3(x^{2}+2x)-10$

$3(x^{2}+2x+1)-3* 1-10$

$3(x+1)^{2}-13$

The vertex of the parabola $y=3(x+1)^{2}-13$ is the point (-1, -13) .

The parabola is concave upward (its quadratic coefficient is positive), so represents the minimum value of . This is our answer.

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Example Question #1 : Graphing Parabolas

What is the vertex of the function $\small f(x)=2(x-4)^2+7$ ? Is it a maximum or minimum?

Possible Answers:

$\small (4,7)$ ; maximum

$\small (-4,7)$ ; maximum

$\small (4,7)$ ; minimum

$\small (-4,7)$ ; minimum

Correct answer:

$\small (4,7)$ ; minimum

Explanation:

The equation of a parabola can be written in vertex form: $\small f(x)=a(x-h)^2+k$ .

The point $\small (h,k)$ in this format is the vertex. If $\small a$ is a postive number the vertex is a minimum, and if $\small a$ is a negative number the vertex is a maximum.

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

In this example, $\small a=2$ . The positive value means the vertex is a minimum.

$\small h=4\ \text{and}\ k=7$

$\small (h,k)=(4,7)$

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Example Question #2 : Graphing Polynomial Functions

Which of the graphs best represents the following function?

$f(x)=\frac{9}{10}x^2-7x+2$

Possible Answers:

None of these

Graph_exponential_

Graph_line_

Graph_parabola_

Graph_cube_

Correct answer:

Graph_parabola_

Explanation:

$f(x)=\frac{9}{10}x^2-7x+2$

The highest exponent of the variable term is two ( x^2 ). This tells that this function is quadratic, meaning that it is a parabola.

The graph below will be the answer, as it shows a parabolic curve.

Graph_parabola_

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Example Question #3321 : Algebra 1

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

Possible Answers:

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

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

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

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

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

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 #2 : Understand Linear And Nonlinear Functions: Ccss.Math.Content.8.F.A.3

Possible Answers:

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

None of the above

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

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

$y = x^{2}$

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 #1 : How To Graph A Quadratic Function

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

Possible Answers:

Graph4

Graph2

Graph

Graph1

Graph3

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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Algebra 1 : How to graph a quadratic function

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #1 : Quadratic Functions

Example Question #1 : Graphing Parabolas

Example Question #2 : Graphing Polynomial Functions

Example Question #3321 : Algebra 1

Example Question #2 : Understand Linear And Nonlinear Functions: Ccss.Math.Content.8.F.A.3

Example Question #1 : How To Graph A Quadratic Function

All Algebra 1 Resources

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