Problem 76 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(x=y^{2}-4\)

Step-by-Step Solution

Answer

The graph of the function is a parabola that opens to the right and intersects the x-axis at \((-4, 0)\). It has no y-intercepts or symmetry.

Step 1: Plot Points

Start by plotting a range of points. Choose several values for \(y\), then substitute those into the equation to find the corresponding \(x\) values. This will give a list of coordinates \((x, y)\) that can be used to plot the points on the graph.

Step 2: Identify Intercepts

An intercept is a point where the graph intersects the x or y axis. For the x-intercept, set \(y=0\) and solve for \(x\). For the y-intercept, set \(x=0\) and solve for \(y\). In this case, because this equation is not defined for \(x<0\), it does not have any y-intercepts. As for the x-intercepts, when setting \(y=0\), the result \(x=-4\) shows that it intersects the x-axis at point \((-4, 0)\).

Step 3: Test for Symmetry

To test for symmetry, one must replace \(x\) by \(-x\) and \(y\) by \(-y\) in the equation. If the original equation remains unchanged, it's symmetrical with respect to the origin. If only \(x\) is replaced and the equation remains unchanged, it's symmetrical respect to the y-axis. Likewise, if replacing only \(y\) leaves the equation unchanged, it's symmetrical respect to the x-axis. Here, the equation does not remain unchanged when \(x\), \(y\), or both are replaced by their negatives, which signifies the graph has no symmetry.

Key Concepts

Plotting Points

Graphing quadratic equations involves creating a visual representation of the equation on a coordinate plane. To start plotting points, we choose a range of values for one variable and solve the equation to find the corresponding values of the other variable. For the equation x=y^2-4, we can pick values such as y = -3, -2, -1, 0, 1, 2, 3, and calculate the respective x values. For example, if y = 2, then x = 2^2 - 4 = 0. This yields the point (0, 2) on the graph.

By plotting several such points, we can begin to see the shape of the graph. Connect the dots in a smooth curve to visualize the parabola described by the equation. It's important to plot enough points to clearly show the curve's direction and width, ensuring that key features of the graph, like the vertex and intercepts, are accurately represented. This visual guide helps to grasp the concept of the function's behavior.

X-intercept and Y-intercept

Intercepts are the points where the graph crosses the axes. The x-intercept is found by setting y = 0 in the equation and solving for x, which gives the point where the graph crosses the x-axis. For our equation, when y = 0, we find that x = -4. Therefore, the graph has a single x-intercept at (-4, 0).

In contrast, a y-intercept occurs where the graph crosses the y-axis, typically found by setting x = 0 in the equation. However, for the equation x = y^2 - 4, the graph does not have any y-intercepts because the value of x cannot be negative. Hence, the graph does not cross the y-axis at any point. This understanding is crucial for accurately sketching the graph and identifying its key components.

Testing for Symmetry

Symmetry of a graph can provide insights into its nature and help to simplify the graphing process. To test for symmetry, we replace x with -x and y with -y and see if we obtain the original equation. If the equation remains unchanged when both variables are replaced, then the graph is symmetric about the origin. If only x is replaced and the equation remains the same, then it is symmetric about the y-axis, and if only y is replaced and the equation remains the same, then it is symmetric about the x-axis.

For our equation, x = y^2 - 4, replacing y with -y does not yield the original equation, indicating there is no symmetry about the x-axis. Similarly, replacing x with -x does not maintain the original equation, which suggests that there is also no symmetry about the y-axis or about the origin. Understanding symmetry helps determine the behavior of the graph and allows a more efficient plot by reflecting points across the axis of symmetry.