Problem 18 - Precalculus with Limits

Question

In Exercises 15-18, complete the table. Use the resulting solution points to sketch the graph of the equation. \( y = 5 - x^2 \)

Step-by-Step Solution

Answer

The y-values for the five chosen x-points are 1, 4, 5, 4, and 1, respectively. Plotting these points on a graph will give a downward facing parabolic curve.

Step 1: Determine Function Values

To fill the table, choose various x-values (both positive and negative) and plug them into the function \( y = 5 - x^2 \) to calculate the corresponding y-values.

Step 2: Calculate the Y values

Let's say the selected x-values are -2, -1, 0, 1, and 2. Plugging these values into the function will yield the following results: when x is -2, y is \(5 - (-2)^2 = 5 - 4 = 1\). When x is -1, y is \(5 - (-1)^2 = 5 - 1 = 4\). When x is 0, y is \(5 - 0^2 = 5\). When x is 1, y is \(5 - 1^2 = 5 - 1 = 4\). Lastly, when x is 2, y is \(5 - 2^2 = 5 - 4 = 1\).

Step 3: Plot the Graph

Now, with these ordered pair coordinates, (-2,1), (-1,4), (0,5), (1,4), (2,1), plot the points on the graph. Since quadratic functions typically form a parabolic curve, these points will lie in a curved fashion. Connect the dots in a curvy pattern to complete the graph.

Key Concepts

Quadratic Functions

A quadratic function is a type of polynomial function that can be generally expressed as \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and the highest degree of the variable \( x \) is two. In the case of \( y = 5 - x^2 \), we can see that it is a quadratic function, where \( a = -1 \), \( b = 0 \), and \( c = 5 \).

One important feature of a quadratic function is its graph, which is a curve called a parabola. Depending on the sign of the coefficient \( a \), this parabola opens upwards (if \( a > 0 \)) or downwards (if \( a < 0 \)). In our example, since \( a \) is negative, the parabola will open downwards, representing a situation where the function has a maximum value at its vertex.

The vertex is the highest or lowest point on the graph, depending on whether it opens up or down. Given the standard form of \( y = a(x-h)^2 + k \), the vertex \( (h, k) \) can be found, but in a more simple form like \( y = 5 - x^2 \), the vertex is at (0, 5) since there is no \( h \) and \( k \) substituted in the equation, and \( x^2 \) is symmetrical about the y-axis.

To sketch a quadratic graph accurately, one must determine several points on the graph using chosen x-values and computing the corresponding y-values, leading us to the next concept: function values calculation.

Graphing Parabolas

Graphing parabolas involves plotting points that satisfy the quadratic function and then connecting these points in a smooth, continuous curve. When tackling \( y = 5 - x^2 \), because of the symmetry of the parabola around its vertex, one can select x-values symmetrically around the vertex, which is at \( x = 0 \) in this case.

Before plotting, create a table of values wherein for each chosen x-value, the corresponding y-value is computed. For a balanced graph, it's best to choose an equal number of positive and negative x-values, as was done with -2, -1, 0, 1, and 2 in our example. Since quadratic functions are symmetrical, positive and negative inputs equidistant from zero produce the same y-value, which explains why \( (-2,1) \) and \( (2,1) \), \( (-1,4) \) and \( (1,4) \) are symmetrical points.

Plotting Steps:

• Begin at the vertex (0,5).
• Move one unit left to (-1,4) and one unit right to (1,4).
• Move two units left to (-2,1) and two units right to (2,1).

After plotting these points, draw a smooth curve through them, ensuring that the curve is symmetrical and opens downwards, illustrating the shape of a

Function Values Calculation

Calculating the function values for a quadratic equation is essential to plot its graph. To compute these values, you'll select points for \( x \) and substitute them into the quadratic equation to find the corresponding \( y \) values. Following this process analytically ensures the accuracy of the graph and helps to identify the shape of the parabola more clearly.

In the exercise, we selected x-values around the vertex: -2, -1, 0, 1, and 2. The chosen x-values were then substituted into \( y = 5 - x^2 \) to find the y-values. For example:

• When \( x = -2 \), \( y = 5 - (-2)^2 = 5 - 4 = 1 \).
• When \( x = -1 \), \( y = 5 - (-1)^2 = 5 - 1 = 4 \).
• When \( x = 0 \), \( y = 5 - 0^2 = 5 \).
• When \( x = 1 \), \( y = 5 - 1^2 = 4 \).
• When \( x = 2 \), \( y = 5 - 2^2 = 1 \).

Once you have a set of points, these can be plotted on the coordinate plane to visualize the parabola's shape. As you plot more points, the graph's accuracy increases, giving you a clearer picture of the quadratic function's behavior. Function value calculation not only aids in plotting but also deepens the understanding of how each input \( x \) affects the output \( y \), which is fundamental when learning about quadratic functions and their graphs.