How To Complete A Function Table | Algebra

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Copyright Algebra 2 Skills Practice

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• 00:04 How to complete a…
• 03:03 How to complete a…

Jump to a specific example Melissa Reese

• Melissa Reese

  Melissa is a former high school math teacher with 9 years of classroom experience in grades 9-12. She has a Bachelor's of Science in Math Education from North Georgia College and State University. She holds a professional teaching license in math grades 6-12, and is also gifted in-field certified.

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Example Solutions Practice Questions

Completing a Function Table

Step 1: Identify the quadratic function. It is in the form of {eq}f(x)=ax^2+bx+c. {/eq}

Step 2: Determine the vertex of our quadratic function, that is, {eq}(-\frac{b}{2a},f(-\frac{b}{2a})) {/eq}.

Step 3: Create a table where the left column has the x-values, and the right column has the y-values. Choose the vertex, at least two points to the right of our vertex, and at least two points to the left of our vertex in our table. You can simply choose the x-values and then plug them into our function to solve for the y-values.

Step 4: Plot these points on our coordinate plane.

Step 5: Connect our points, and extend our line beyond the point with arrows to complete drawing our quadratic function.

Completing a Function Table Vocabulary

Function: A function is a rule that assigns a single output to each allowable input. We typically denote our inputs by {eq}x {/eq} and our outputs by either {eq}y {/eq} or {eq}f(x) {/eq}.

We can think of functions like machines, where:

• Our inputs are our raw materials being put into the machine
• Our function is the machine itself that converts these raw materials into finished products
• Our outputs are the finished products that come out of the machine

Quadratic function: Quadratic functions are of the form {eq}f(x)=ax^2+bx+c, {/eq}, where {eq}a, \ b, \ c, {/eq} are numbers and {eq}a\neq0 {/eq}. A special property about quadratic functions is that they have a constant "change in the change," that is, the difference between the consecutive differences in y-values are constant.

Quadratic functions are U-shaped and symmetric in nature, where they look identical on either side of their vertex (the highest or lowest point on a quadratic function). Our vertex will always be the point {eq}(-\frac{b}{2a},f(-\frac{b}{2a})). {/eq} This will be our lowest point if {eq}a>0 {/eq}, and this will be our highest point if {eq}a<0. {/eq}

Table/graph of a quadratic function: When we are given a quadratic function, we can construct a chart of points to eventually plot on a coordinate plane (a plane, together with two perpendicular lines called axes that meet at the location of 0, called the origin, on each number line). The left-hand column will have the x-values, and the right-hand column will plug these x-values into our quadratic function to determine our y-values. Typically, we plug in our vertex, two points to the left of our vertex, and two points to the right of our vertex. Below is an example of what such a table will look like:

{eq}\begin{array}{|c|c|} \hline \mathbf{x} & \mathbf{f(x)=x^2}\\ \hline -2 & f(-2)=4\\ -1 & f(-1)=1 \\ 0 & f(0)=0\\ 1 & f(1)=1\\ 2 & f(2)=4\\ \hline \end{array} {/eq}

Given a table of values, we can plot all of the points and connect them to graph our quadratic function. Since our equation extends beyond these points, we will extend the line with arrows at the end to indicate the line goes on to {eq}\pm\infty. {/eq} An example of such a quadratic function is given below:

We will now demonstrate two examples in detail.

How to Complete a Function Table: Example 1

Graph {eq}f(x)=x^2 - 6x + 7 {/eq}.

Step 1: Identify the quadratic function in question.

Our quadratic function is {eq}f(x)=x^2 - 6x + 7 {/eq}, where {eq}a=1, \ b=-6, \text{ and } c=7. {/eq}

Step 2: Determine the vertex of our quadratic function.

The x-value of our vertex will be {eq}x=-\frac{-6}{2\times(1)}=3. {/eq} Plugging this into our equation, we get {eq}\begin{align} f(3)=3^2-6(3)+7=9-18+7=-2 \end{align} {/eq}

Thus, our vertex is {eq}(3,-2). {/eq}

Step 3: Create a table.

Let's choose two points to the left of {eq}x=3, {/eq} say {eq}x=1,2, {/eq} and two points to the right of {eq}x=3, {/eq} say {eq}x=4,5. {/eq} Plugging all of these points into our function, we get the following table:

{eq}\begin{array}{|c|c|} \hline \mathbf{x} & \mathbf{f(x)=x^2-6x+7}\\ \hline 1 & f(1)=2\\ 2 & f(2)=-1 \\ 3 & f(3)=-2\\ 4 & f(4)=-1\\ 5 & f(5)=2\\ \hline \end{array} {/eq}

Step 4: Plot these points on our coordinate plane.

We can adjust our plane so that we can see our points easily. We can then plot them on the coordinate plane:

Step 5: Connect our points.'

Our final quadratic function will look like the following:

How to Complete a Function Table: Example 2

Graph {eq}f(x)=-4x^2 - 12x + 21 {/eq}.

Step 1: Identify the quadratic function in question.

Our quadratic function is {eq}f(x)=-4x^2 - 12x + 21 {/eq}, where {eq}a=-4, \ b=-12, \text{ and } c=21. {/eq}.

Step 2: Determine the vertex of our quadratic function.

The x-value of our vertex will be {eq}x=-\frac{-12}{2\times(-4)}=-\frac{-12}{-8}=-\frac32. {/eq} Plugging this into our equation, we get {eq}\begin{align} f\big(-\frac32\big)=-4\big(-\frac32\big)^2-12\big(-\frac32\big)+21&=-4\big(\frac94\big)+\frac{36}{2}+21\\ &=-9+18+21\\ &=30 \end{align} {/eq}

Thus, our vertex is {eq}(-\frac32,30). {/eq}

Step 3: Create a table.

Let's choose two points to the left of {eq}x=-\frac32, {/eq} say {eq}x=-2,-\frac52, {/eq} and two points to the right of {eq}x=-\frac32, {/eq} say {eq}x=-1,-\frac12. {/eq} Plugging all of these points into our function, we get the following table:

{eq}\begin{array}{|c|c|} \hline \mathbf{x} & \mathbf{f(x)=-4x^2-12x+21}\\ \hline -\frac52 & f\big(-\frac52\big)=26\\ -2 & f(-2)=29 \\ -\frac32 & f\big(-\frac32\big)=30\\ -1 & f(-1)=29\\ -\frac12 & f\big(-\frac12\big)=26\\ \hline \end{array} {/eq}

Step 4: Plot these points on our coordinate plane.

We can adjust our plane so that we can see our points easily. We can then plot them on the coordinate plane:

Step 5: Connect our points.

Our final quadratic function will look like the following:

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