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.
yequals
minus(xplus 2)squarednegative 4
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Part 1
Complete the table of values for the function.
x
y
negative 4
enter your response here
negative 3
enter your response here
negative 2
enter your response here
negative 1
enter your response here
0
enter your response here
The given function appears to be \( y = -(x + 2)^2 - 4 \). This is a quadratic function because it has \( x \) raised to the second power. To illustrate why this is not a linear function, let's fill in the table of values and then sketch the graph.
First, complete the table for the given x values:
- When \( x = -4 \), \( y = -((-4) + 2)^2 - 4 = -(2)^2 - 4 = -4 - 4 = -8 \).
- When \( x = -3 \), \( y = -((-3) + 2)^2 - 4 = -(-1)^2 - 4 = -1 - 4 = -5 \).
- When \( x = -2 \), \( y = -((-2) + 2)^2 - 4 = -(0)^2 - 4 = -4 \).
- When \( x = -1 \), \( y = -((-1) + 2)^2 - 4 = -(1)^2 - 4 = -1 - 4 = -5 \).
- When \( x = 0 \), \( y = -((0) + 2)^2 - 4 = -(2)^2 - 4 = -4 - 4 = -8 \).
So the table with values will look like this:
\[
\begin{array}{cc}
\text{x} & \text{y} \\
\hline
-4 & -8 \\
-3 & -5 \\
-2 & -4 \\
-1 & -5 \\
0 & -8 \\
\end{array}
\]
Now, let's sketch the graph to visualize why it's a nonlinear function (please draw this out on paper with a pencil):
1. Plot the points from the table on a graph with the x-axis (horizontal) and y-axis (vertical).
2. Connect the dots.
You'll notice that the points create a parabola, which opens downward due to the negative sign in front of the squared term. This confirms that the function is indeed nonlinear.
For a function to be linear, it must have the form \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept. The graph of a linear function is a straight line, and the rate of change (the slope) is constant. This is not the case for the function given.
To find a linear function, let's choose a simple one:
\( y = 2x + 1 \)
Now complete the table for this linear function:
- When \( x = -4 \), \( y = 2(-4) + 1 = -7 \).
- When \( x = -3 \), \( y = 2(-3) + 1 = -5 \).
- When \( x = -2 \), \( y = 2(-2) + 1 = -3 \).
- When \( x = -1 \), \( y = 2(-1) + 1 = -1 \).
- When \( x = 0 \), \( y = 2(0) + 1 = 1 \).
The graph of this function would be a straight line with a slope of 2 and a y-intercept of 1. The table and graph demonstrate that the rate of change between \( x \) values is constant, which characterizes a linear function.