Problem 26 - College Algebra and Calculus: An Applied Approach

Question

Complete the table. Use the resulting solution points to sketch the graph of the equation. \(y=5-x^{2}\) $$\begin{array}{|l|l|l|l|l|l|}\hline x & -2 & -1 & 0 & 1 & 2 \\\\\hline y & & & & & \\ \hline\end{array}$$

Step-by-Step Solution

Answer

By substituting the x-values into the equation \(y = 5 - x^{2}\), the y-values corresponding to -2, -1, 0, 1, 2 are 1, 4, 5, 4, 1 respectively. These points form a parabolic curve when sketched on a graph.

Step 1: Table completion

Enter each given x-value into the given equation to determine the y-value. The equation is \(y=5-x^{2}\). So, for each value of 'x', calculate the square of 'x', subtract it from 5 and record this as the y-value.

Step 2: Calculation for x=-2

Substitute x=-2 into the equation. Therefore, \(y = 5- (-2)^2 = 5 -4 = 1\)

Step 3: Calculation for x=-1

Substitute x=-1 into the equation. Therefore, \(y = 5 - (-1)^2 = 5 - 1 = 4\)

Step 4: Calculation for x=0

Substitute x=0 into the equation. Therefore, \(y = 5 - 0^2 = 5 - 0 = 5\)

Step 5: Calculation for x=1

Substitute x=1 into the equation. Therefore \(y = 5 - 1^2 = 5 - 1 = 4\)

Step 6: Calculation for x=2

Substitute x=2 into the equation. Therefore, \(y = 5 - 2^2 = 5 - 4 = 1\)

Step 7: Sketch a graph

Now, plot these values on a graph where 'x' is on the x-axis and 'y' is on the y-axis. You will be able to see the curve shape of the equation.

Key Concepts

Parabolas

A parabola is a curved, symmetrical shape that is the graph of a quadratic function. In a quadratic equation like \( y = 5 - x^2 \), the graph produces a parabola. This specific equation graphed forms an "upside-down" parabola because the coefficient of \( x^2 \) is negative (-1).

Key properties of parabolas include:

• Vertex: The highest or lowest point. Here, the vertex is at the highest point since it's an upside-down parabola. For \( y = 5 - x^2 \), the vertex is at (0, 5).
• Axis of Symmetry: A vertical line through the vertex that divides the parabola into two symmetrical halves. For this equation, the axis of symmetry is \( x = 0 \).
• Direction: The parabola opens downward due to the negative sign in front of \( x^2 \).

Understanding these properties helps visualize how the parabolic shape will look when you graph it.

Completing Tables

Completing a table for a quadratic function involves plugging in \(x\) values to determine their corresponding \(y\) values. This is done using the given equation.

Start by taking each \(x\) value from the table. For instance, when \(x = -2\):

• Substitute \(x = -2\) into \(y = 5 - x^2\) which gives \(y = 5 - (-2)^2 = 1\).

Repeat this calculation for the other values like \(x = -1, 0, 1,\) and \(2\). The complete table will give you a series of coordinate points: \((-2, 1), (-1, 4), (0, 5), (1, 4), (2, 1)\).

These points represent specific locations on the graph. Calculating correctly ensures an accurate graph, visually demonstrating the equation’s behavior.

Graphing Equations

Graphing equations, especially quadratic ones, involves plotting points on a grid to visually represent the function. Start by using the points calculated from the table: \((-2, 1), (-1, 4), (0, 5), (1, 4), (2, 1)\).

Plot each point on the graph where the horizontal line is the x-axis, and the vertical line is the y-axis. Once plotted, these points form the parabola of the quadratic equation.

Connect the points smoothly because quadratic equations graph as a continuous curved line. The symmetry of the parabola means the points on either side of the vertex are mirrored, making it easier to plot accurately.

Graphing provides a visual understanding of how equations translate into shapes, highlighting crucial features like peaks and symmetry.