Problem 14 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the function and compare it with the graph of \(y=x^{2}\) \(y=-x^{2}+2\)

Step-by-Step Solution

Answer

The graph of the function \(y = -x^{2} + 2\) is a downward-opening parabola with the vertex at (0,2). This differs from the graph of \(y = x^{2}\), which is an upward-opening parabola with the vertex at the origin.

Step 1: Graph the function \(y = -x^{2} + 2\)

A good place to start with quadratic functions is with the vertex. This function is in vertex form \(y = a(x - h)^{2} + k\), where \(a = -1\), \(h = 0\), and \(k = 2\). So the vertex is at (0,2). Because the coefficient of \(x^{2}\) is negative, the parabola opens down. Other key points can be found by substituting x-values into the equation, picking values on either side of the vertex ( 0 ). For example, if we substitute \(x = -1\) and \(x = 1\), we get \(y = 1\). Plot these points along with the vertex and draw a parabola that passes through those points.

Step 2: Compare with the graph of \(y = x^{2}\)

Now, recall the basic shape of \(y = x^{2}\). This is a parabola that opens upward with the vertex at the origin (0,0). Comparing this with the function \(y = -x^{2} + 2\), you can see that the main differences are in the direction of the opening and the position of the vertex. The given function's graph has a vertex that is higher (at (0,2) versus (0,0)) and it opens down as opposed to upwards.

Key Concepts

Vertex Form of a Quadratic

Understanding the vertex form of a quadratic equation is essential for graphing parabolas and analyzing their properties. The vertex form of a quadratic is given by \(y = a(x - h)^{2} + k\), where \(a\), \(h\), and \(k\) are constants. The value of \(a\) determines the direction of the parabola's opening: if \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards. The point (\(h\),\(k\)) represents the vertex of the parabola, which is the highest or lowest point on the graph depending on the value of \(a\).

For instance, in the equation \(y = -x^{2} + 2\), the vertex form tells us that the vertex is at (0,2) and since the coefficient of \(x^{2}\) is -1, the parabola opens downwards. This indicates that the point (0,2) is the maximum point on the graph.

By shifting the values of \(h\) and \(k\), the vertex moves accordingly on the coordinate plane, which allows students to graph various parabolas without needing to calculate additional points unless further graph detail is necessary.

Graphing Parabolas

Graphing parabolas is a visual way to understand the behavior of quadratic functions. To graph a parabola from its vertex form, first identify the vertex and plot it on the coordinate plane. Next, determine the direction of the parabola's opening based on the coefficient of \(x^{2}\). If the coefficient is positive, the parabola opens upwards; otherwise, it opens downwards.

Once the vertex is plotted, the next step involves finding additional points to define the shape of the parabola. Choose x-values to the left and right of the vertex's x-coordinate, substituting these into the equation to find the corresponding y-values. Plot these points and draw a smooth curve through all the points, making sure it reflects the correct opening direction of the parabola.

In our exercise with \(y=-x^{2}+2\), after plotting the vertex, we chose \(x = -1\) and \(x = 1\) to find more points. Both give us the y-value of 1, which we then plot. Drawing a curve through these points reveals the parabola's symmetry and illustrates the downward opening as predicted.

Comparing Quadratic Graphs

Comparing different quadratic graphs helps students understand the effects of changes in the equation on the graph's shape and position. Quadratics in the form \(y = x^{2}\) have a basic parabola shape with a vertex at the origin and an upward opening. This serves as a reference point for comparison.

When looking at the exercise's graph \(y = -x^{2} + 2\) versus \(y = x^{2}\), two main differences are evident. First, the direction of opening is opposite because of the negative coefficient of \(x^{2}\) in the exercise's equation. Second, the vertex is not at the origin but instead is higher up at (0,2). Knowing these differences, we can predict how alterations in equations will affect their graphs.

For example, changing the coefficient of \(x^{2}\) stretches or compresses the parabola, while altering \(h\) and \(k\) shifts the vertex around the plane. For in-depth comparison, analyzing the effect of each component—\(a\), \(h\), and \(k\)—on the graph's shape, size, and position enhances a student's ability to graph and understand quadratics with confidence.