Problem 6 - Technical Mathematics with Calculus

Question

For each equation make a table of point pairs, taking integer values of \(x\) from -3 to 3, plot these points, and connect them with a smooth curve. $$y=4-2 x^{2}$$

Step-by-Step Solution

Answer

Calculated points for the graph are: (-3, -14), (-2, -4), (-1, 2), (0, 4), (1, 2), (2, -4), (3, -14). The graph is a downward-opening parabola.

Step 1: Create the table

To plot the given quadratic equation, start by setting up a table with two columns: one for integer values of the independent variable 'x' ranging from -3 to 3, and the second column for the corresponding values of the dependent variable 'y'. Calculate 'y' using the equation 'y = 4 - 2x^2'. Fill in the table with the results.

Step 2: Calculate y-values

For each integer value of x from -3 to 3, substitute the value into the equation to find the corresponding y-value. For instance, if x = -3, compute y = 4 - 2(-3)^2 = -14. Perform this calculation for each value of x.

Step 3: Plot the points

On graph paper, plot each point pair ('x', 'y') from the table. Ensure the points are accurately represented on the Cartesian coordinate system.

Step 4: Connect the points with a smooth curve

After plotting all the points, connect them with a smooth, parabolic curve. The curve should open downwards because the coefficient of the 'x^2' term is negative in the equation.

Key Concepts

Graphing Quadratic Functions

Understanding how to graph quadratic functions is a fundamental skill in algebra. A quadratic function can be identified by its standard form, which is written as y = ax^2 + bx + c. When graphing such a function, you'll notice the resulting shape is called a 'parabola'.

Let's put this into practice by considering the equation y = 4 - 2x^2. The process begins by identifying a range of x-values; in this case, we take integer values from -3 to 3. For each selected x-value, you calculate the corresponding y-value. Once you have a set of point pairs, you can plot these points on a graph. Start by placing a dot on the coordinate that corresponds to each point pair.

After all points are plotted, you will see that they form a pattern. Join these points with a smooth curve, and you have your graph. Graphing can reveal the nature of the function, like its concavity and the location of its vertex, which is the highest or lowest point of the parabola, depending on whether it opens upwards or downwards.

Creating Tables for Equations

When graphing functions, creating a table of values is an effective first step. This approach organizes the input and output of a function, which in turn simplifies the plotting process. To create a table for y = 4 - 2x^2, start with two columns, one for x-values and one for y-values.

Since we're dealing with integer x-values from -3 to 3, list these in the first column. Next, for each x-value, calculate the corresponding y-value using the quadratic equation. This will populate the second column. It's crucial to calculate each y-value accurately to ensure a correct graph.

For example, when x is -3, we plug it into the equation to get y = 4 - 2(-3)^2 = -14. Repeat this process for each x-value. The resulting table facilitates a clear visualization of how y-values change with x-values which lays the groundwork for plotting them on a graph.

Parabolic Curves

Parabolic curves are the graphical representation of quadratic functions and possess unique properties that make them fascinating in the realm of mathematics and physics. In a quadratic function like y = 4 - 2x^2, the sign in front of x^2 determines the direction in which the parabola opens.

In our example, because the coefficient of x^2 is negative, the parabola opens downwards. Each parabola has a vertex, the peak or trough of the curve. For y = 4 - 2x^2, the vertex is at the point (0, 4), where x is zero. This point is the maximum value of the function, as the curve extends indefinitely downwards from there.

The axis of symmetry is another important feature of parabolic curves. It is a vertical line that passes through the vertex, dividing the parabola into mirror images. For a downward-opening parabola like ours, you can imagine throwing a ball into the air; it rises until reaching a maximum height (the vertex) and then falls back down in a symmetrical path.