Question

Graph the function and make a table of the X and Y values.

Answer 1

`\begin{array}c\cline{1-2}x&y\\\cline{1-2}-1&6\\\cline{1-2}0&0\\\cline{1-2}1&-2\\\cline{1-2}2&0\\\cline{1-2}3&6\\\cline{1-2}\end{array}`

See the attachment for the graph of the function.

Explanation:

The function f(x) = 2(x - 1)² - 2 is a quadratic function in vertex form.

The vertex form of a quadratic function is:

`\boxed{f(x) = a(x - h)^2 + k}`

where:

• a is the leading coefficient that determines the direction and steepness of the parabola. If a < 0 the parabola opens downward, and if a > 0 the parabola opens upward.
• (h, k) is the vertex of the parabola.

Therefore, for the function f(x) = 2(x - 1)² - 2:

• The leading coefficient is positive, so the parabola opens upward.
• The vertex is (1, -2).

To create a table of x and y values for the given function, choose a range of x values that corresponds to the given coordinate plane, calculate the corresponding y values using the function, and then list them in a table:

`\begin{array}\cline{1-3}x&f(x)&y\\\cline{1-3}-1&2(-1-1)^2-2&6\\\cline{1-3}0&2(0-1)^2-2&0\\\cline{1-3}1&2(1-1)^2-2&-2\\\cline{1-3}2&2(2-1)^2-2&0\\\cline{1-3}3&2(3-1)^2-2&6\\\cline{1-3}\end{array}`

To graph the function:

• Plot the calculated points from the x and y table.
• Draw a smooth curve through the points that is symmetric about the x-value of the vertex, x = 1.

Answer 2

See below

Explanation:

f(x) = 2(x-1)²-2

To graph the function and create a table of x and y values, we can choose a range of x values and calculate the corresponding y values.

Let's choose a range of x values from -2 to 4.

Table:

`\begin{array}c x & f(x) \\ -2 & 16 \\ -1 & 6 \\ 0 & 0 \\1 & -2 \\ 2 & 0 \\ 3 & 6\\ 4 & 16 \\ \end{array}`

For Graph : See Attachment

In the graph, We can see the parabolic shape of the function f(x).

The vertex of the parabola is at (1, -2), and the parabola opens upward.

The y values in the table correspond to the heights of the points on the graph.