Make a table for different values of x as shown below.

Draw a graph using the tabulated values as shown below.

So, Graph 1 represents the function y = - 12| x | + 1.

Choose values for x.

Plot the points on a graph.

The above graph matches with the Graph 2.

So, Graph 2 is the correct answer.

y = - 2x3 + 2

3y = - 2x + 6

Choose values for x

Plot the points on a graph.

The above graph matches with the graph in choice B.

So, choice B is the correct answer.

The x-coordinate of the vertex = - b2a = 02(-13) = 0

The values of y = - (x23) + 2 for the x-values to the left and right of x = 0 are:

The graph 2 satisfies the above table.

The absolute value function is f(x) = | x |

Graph 3 represents the absolute value function.

The x-coordinate of the vertex = - b2a = - 02(- 1) = 0

The values of y = - x² + 3 for the x-values to the left and right of x = 0 are tabulated below:

Plot the points and join them with a smooth curve as shown.

The graph matches with Graph 3.

From the graph (0,0), (1,4), and (4,8) lies on the curve.

Substitute the points in the given equations and verify.

y = 4(0) - 1 = - 1 ≠ 0
So, y = 4(x) - 1 does not represent the graph.

y = 4(0)+ 2 = 2 ≠ 0
So, y = 4(x) + 2 does not represent the graph.

y = 3(0) = 0
y = 3(1) = 3≠ 4
So, y = 3(x) does not represent the graph.

y = 4(0) = 0
y = 4(1) = 4
y = 4(4) = 8. So, y = 4(x) equation represent the graph.

A step function is a special type of function whose graph is a series of line segments.

Among the graphs, Graph 3 has a series of line segments.

So, Graph 1, Graph 2 and Graph 4 does not represents a step function.

The given graph is in two pieces.

The slope of the line, left to the origin is - 23 and the y-intercept = 0.

The equation of the line in the slope intercept form is given by y = mx + b.

Substituting the values for m = - 23, b = 0, we get the line equation as,
y = - 23x.

The slope of the line to the right of the origin is 0 and the y-intercept is 4.

Substitute the values of m = 0 and b = 4 in the equation y = mx + b to get,
y = 4.

Therefore, the graph represents the function,
y = - 23x if x ≤ 0
y = 4 if x > 0

y = 2x + 8

From the equation y = 2x + 8, slope is 2 and y-intercept is 8.

Plot the point (0, b) when b = 8.

Use the slope to locate a second point on line.

m = 21 = riserun = move 2 unit upmove 1 units right

Draw a line through the two points.

The graph of the equation y = 2x + 8 matches with Graph 3.

Make a table of values that includes both positive and negative x values.

x	- 3	- 2	- 1	0	1	2	3
y	- 7	- 6	- 5	- 4	- 3	- 2	- 1

Graph the ordered pairs in the table shown and connect the points with a smooth curve.

Then the graph obtained matches with Graph 2.

So, Graph 2 represents the equation y = - |x - 4|.

f(x) = |x| + 2 when x < 1
                        x² + x when x ≥ 1

|x| + 2 is an absolute value function.

When x < 1, the value of the function can be any real number.

x² + x is a quadratic equation.

When x ≥ 1, the value of the function is greater than or equal to 2.

So, we can observe that Graph 2 represents the given relation.

Identify the graph for the function shown.
f(x) = - 2 if x ≤ 0
            x - 1 if 0 < x ≤ 3
            x + 1 if x > 3

When x ≤ 0, the value of the function remains constant at - 2.

When 0 < x ≤ 3, the value of the function is (- 1, 2].

When x > 3, the value of the function is (4, ∞).

So, we can observe that the Graph 1 represents the given function.

Make a table of values from the given graph that includes both positive and negative x values.

x	- 3	- 2	- 1	1	2	3
y	9	6	3	3	6	9

From the given choices, y = |3x | satisfies the values given in above table.

So, the equation y = |3x | represents the given graph.