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.
y = - (x - 3) ^ 2 + 3
Complete the table of values for the function.
x 1 2 3 4 5
y
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.
To complete the table of values for the function y = -(x - 3)^2 + 3, we substitute the given values of x into the equation and evaluate for y.
x | y
1 | -5
2 | -3
3 | 0
4 | -3
5 | -5
Now let's plot these values on a graph to determine if the function is linear or nonlinear.
The graph should have x-values ranging from 1 to 5 and y-values ranging from -5 to 0. The point (1, -5) will be the lowest point on the graph since the expression -(x - 3)^2 + 3 is a downward-facing quadratic.
Here is a rough sketch of the graph:
|
|
-5 | *
|
| *
|
-3 | *
|
|
0 | * * * * *
|
|
-----------------------------------------------------
1 2 3 4 5
From the graph and the table, we can see that the function is not linear; it is a quadratic function. The points do not form a straight line, and the graph is a downward-facing parabola.
To find a linear function, we need a set of points that form a straight line. Since the given function is nonlinear, we cannot find a linear function that perfectly fits these points.
To determine whether the function is linear or nonlinear, we can look at both the table of values and the graph.
To complete the table of values for the given function y = -(x - 3)^2 + 3, we substitute the given x-values and evaluate for y:
x | y
1 | 1
2 | -1
3 | 3
4 | -1
5 | 1
Now, let's plot these points on a graph:
- Here, x is the independent variable, which means it is plotted on the x-axis.
- The corresponding y-values are the dependent variable and are plotted on the y-axis.
Now, let's connect these points on the graph using a straight line:
(Note: Although the function is nonlinear, it is beneficial to see the line connecting the plotted points for explanation purposes.)
After plotting the points and connecting them with a line, we can observe that the resulting graph forms a "V" shape. This indicates that the function is nonlinear because it does not produce a straight line.
In order to find a linear function, we need to look for a function that produces a straight line when plotted on a graph. A popular form of a linear function is y = mx + b, where m represents the slope and b is the y-intercept.
To find a linear function that matches the table and the graph, we need to observe that the given function, y = -(x - 3)^2 + 3, is a quadratic equation. However, we can create a linear approximation by considering only the vertex point of the quadratic function, which lies on the line of symmetry.
For the given function, the vertex point is (3, 3), which is on the line of symmetry, x = 3. Therefore, we can approximate the given function to y = 3. This simplified linear function is a line with a constant y-value of 3, which means it does not change as x varies.
In summary, the given function y = -(x - 3)^2 + 3 is nonlinear as it forms a "V" shape on the graph. A linear approximation could be y = 3, where y remains constant regardless of the x-value.