Problem 26 - Calculus Volume 1

Question

Set up a table to sketch the graph of each function using the following values: \(x=-3,-2,-1,0,1,2,3\) \(f(x)=-x^{2}\)

Step-by-Step Solution

Answer

The graph is an inverted parabola passing through these points: (-3,-9), (-2,-4), (-1,-1), (0,0), (1,-1), (2,-4), (3,-9).

Step 1: Calculate f(x) for x=-3

To find the value of \(f(x)\) when \(x = -3\), substitute \(-3\) into the equation \(f(x) = -x^2\). This gives:\[f(-3) = -(-3)^2 = -9\]

Step 2: Calculate f(x) for x=-2

Substitute \(x = -2\) into the equation \(f(x) = -x^2\). Calculate as follows:\[f(-2) = -(-2)^2 = -4\]

Step 3: Calculate f(x) for x=-1

Substitute \(x = -1\) into the equation \(f(x) = -x^2\). Calculate as follows:\[f(-1) = -(-1)^2 = -1\]

Step 4: Calculate f(x) for x=0

Substitute \(x = 0\) into the equation \(f(x) = -x^2\). Calculate as follows:\[f(0) = -(0)^2 = 0\]

Step 5: Calculate f(x) for x=1

Substitute \(x = 1\) into the equation \(f(x) = -x^2\). Calculate as follows:\[f(1) = -(1)^2 = -1\]

Step 6: Calculate f(x) for x=2

Substitute \(x = 2\) into the equation \(f(x) = -x^2\). Calculate as follows:\[f(2) = -(2)^2 = -4\]

Step 7: Calculate f(x) for x=3

Substitute \(x = 3\) into the equation \(f(x) = -x^2\). Calculate as follows:\[f(3) = -(3)^2 = -9\]

Step 8: Construct the Table

Now compile all the calculated values into the table as follows: \[\begin{array}{|c|c|}\hlinex & f(x) \\hline-3 & -9 \-2 & -4 \-1 & -1 \0 & 0 \1 & -1 \2 & -4 \3 & -9 \\hline\end{array}\]

Step 9: Sketch the Graph

Using the table values, plot the points \((-3, -9), (-2, -4), (-1, -1), (0, 0), (1, -1), (2, -4), (3, -9)\) on a coordinate plane. Draw a smooth curve through these points; you will see it forms an inverted parabola.

Key Concepts

Graph Sketching

Graph sketching is a visual way of understanding how a mathematical function behaves. It involves plotting points on a graph based on values calculated from a function. For the function given in the problem,

• First, calculate several values of the function for different inputs, as done in the original solution.
• Next, plot these points on a graph where the horizontal axis represents the variable 'x' and the vertical axis represents 'f(x)'.
• Once the points are plotted, smoothly connect them to reveal the shape of the curve.

In this exercise, the curve is a downward opening parabola because the function \(f(x) = -x^2\) is a quadratic function with a negative leading coefficient. This tells us that every value calculated will be negative or zero. Graph sketching helps identify trends and the overall behavior of functions at a glance.

Function Evaluation

Function evaluation is determining the output of a function for a given input. It involves substituting values into the function's formula. To evaluate a function like \(f(x) = -x^2\), simply replace 'x' with a number.

• For example, substitute \(x = -3\) to get \(f(-3) = -(-3)^2 = -9\).
• This process is repeated for each desired value of 'x'.
• The resulting 'f(x)' values are used to help sketch the function's graph.

This simple substitution helps us understand how the function reacts over different numbers, and is particularly useful in plotting functions on a graph and observing their patterns like parabolas.

Coordinate System

The coordinate system is a framework for graphically representing numbers as points on a plane. Typically, it's a two-dimensional plane with an x-axis (horizontal) and y-axis (vertical), crossing at the origin point \((0,0)\).

• When plotting a point, first find the appropriate 'x' position, then move up or down the 'y' axis to reach 'y'.
• For example, the point \((-3, -9)\) means moving 3 units left on the x-axis and 9 units down on the y-axis.

This system allows us to visualize mathematical relationships, as it makes it easy to observe how changes in 'x' affect 'y', which is critical in understanding the shape of the function we work with, like the parabola in this problem.