Problem 16 - Technical Mathematics with Calculus

Question

Rewrite each equation in explicit form and graph for integer values of \(x\) from -3 to 3 . $$y+x^{2}=12$$

Step-by-Step Solution

Answer

The explicit form is \( y = 12 - x^2 \). Plot the points from the table on a graph for integer values of x from -3 to 3 and draw a parabola through the points.

Step 1: Solve the equation for y

To write the equation in explicit form, solve for y. Start with the given equation: \( y + x^2 = 12 \).Subtract \(x^2\) from both sides to isolate y: \( y = 12 - x^2 \).

Step 2: Create a table of values

To graph the equation, create a table of values for y using the explicit form \( y = 12 - x^2 \)for integer values of x from -3 to 3. Calculate the corresponding y values.

Step 3: Plot the points on a graph

Use the (x, y) pairs from the table of values to plot the points on a coordinate plane. Since this is a quadratic equation, the graph will be a parabola.

Step 4: Draw the graph

After plotting the points, draw a smooth curve connecting them to form the parabola. The vertex of the parabola will be at the point where x=0.

Key Concepts

Explicit Form

When dealing with quadratic equations, rewriting them in explicit form allows us to easily identify how the graph will look and how we can plot it. An explicit form is when the equation is solved for one variable, typically 'y', in terms of the other variable, 'x'. For example, in the given exercise, we start with the equation in implicit form,
\( y + x^2 = 12 \)
and manipulate it to isolate 'y'.
\( y = 12 - x^2 \)
In this explicit form, we can see that for every x-value, there is a single y-value. This is the first step in understanding the nature of the graph we are about to draw.

Table of Values

A table of values is a practical tool for organizing and visualizing the relationship between x-values and their corresponding y-values. After finding the explicit form, creating a table helps in plotting the graph accurately. Here, you list integer values of 'x'—in this case from -3 to 3—and calculate the corresponding 'y' using the explicit form (
\( y = 12 - x^2 \)
). By substituting the x-values into the equation, you generate y-values, which will give you coordinate pairs that can be plotted on a graph. The completion of the table makes it easier to spot patterns and anticipate the shape of the graph.

Coordinate Plane Plotting

With the table of values in hand, the next step is to plot these points on a coordinate plane. The coordinate plane is a two-dimensional surface where each point is determined by an x-value (horizontal position) and a y-value (vertical position). Start by drawing a horizontal line for the x-axis and a vertical line for the y-axis that intersect at the origin (0,0). Mark evenly spaced points along each axis, then plot the (x, y) pairs from your table, placing a dot for each pair. Remember to label your axes and choose an appropriate scale to accurately represent the data points.

Parabola Graph Characteristics

The graph of a quadratic equation is always a parabola. A parabola has certain characteristics, such as its direction (opening up or down), vertex (the highest or lowest point), and axis of symmetry (a vertical line through the vertex splitting the parabola into mirror images). For the equation
\( y = 12 - x^2 \)
the parabola opens downwards because the coefficient of
\( x^2 \)
is negative. The vertex here is (0, 12) since the y-value is at its maximum when x is 0. The axis of symmetry is the line
\( x = 0 \)
which is simply the y-axis in this case. By understanding these characteristics, you can sketch the parabola more quickly and accurately after plotting the initial points.