Challenge Decide whether the function is linear or nonlinear using both a table and a graph. Use pencil and paper. Find a linear function. Make a table and a graph for the function to explain why it is linear.

Question

y = - (x - 3) ^ 2 + 3

Complete the table of values for the function.

x 1 2 3 4 5
y

Answer 1

Using the given function, we can substitute the values of x into the function to find the corresponding values of y.

x 1 2 3 4 5
y -9 -7 3 -7 -19

Now, let's graph the function:

Plotting the points (1, -9), (2, -7), (3, 3), (4, -7), and (5, -19) on a graph, we can connect them with a line.

Since the graph of the function does not form a straight line, we can conclude that the given function is nonlinear.

To find a linear function, we need a straight line. One way to find a linear function is to choose two points on the graph and use them to find the slope (m) of the line.

Let's choose the points (1, -9) and (4, -7) from the graph.

m = (change in y) / (change in x)
= (-7 - (-9)) / (4 - 1)
= 2 / 3

We now have the slope of the line. We can use the point-slope form of a linear equation to find the equation of the line.

y - y1 = m(x - x1)

Using the point (1, -9), we have:
y - (-9) = (2/3)(x - 1)
y + 9 = (2/3)(x - 1)

Simplifying the equation, we have:
y + 9 = (2/3)x - 2/3

Subtracting 9 from both sides of the equation, we have:
y = (2/3)x - 2/3 - 9
y = (2/3)x - 2/3 - 27/3
y = (2/3)x - 29/3

This equation represents a linear function.

Let's complete the table of values for the linear function:

x 1 2 3 4 5
y -29/3 -25/3 -21/3 -17/3 -13/3

Plotting these points on a graph and connecting them with a straight line, we can see that the graph forms a straight line, confirming that the function is linear.