Problem 75 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(x=y^{2}-1\)

Step-by-Step Solution

Answer

The graph of \(x=y^{2}-1\) is a U-shaped curve that opens towards the positive x-axis, is symmetrical with respect to the y-axis and passes through the x-intercept \(-1\). However, there are no y-intercepts.

Step 1: Find the x-intercept

Set \(y=0\) in the equation \(x=y^{2}-1\) and solve for \(x\). This yields \(x=(-0)^{2}-1=-1\). Therefore, the x-intercept is \(-1\).

Step 2: Find the y-intercept

Set \(x=0\) in the equation \(x=y^{2}-1\) and solve for \(y\). This doesn't result in realistic values for \(y\), which means that there are no y-intercepts.

Step 3: Check for symmetry

Replace \(y\) with \(-y\) in the equation \(x=y^{2}-1\). This yields \(x=(-y)^{2}-1=y^{2}-1\), which is the same as the original equation. Thus, this function is symmetric with respect to the y-axis.

Step 4: Sketch the graph

Based on the results from steps 1-3, first plot the x-intercept on the x-axis. Since there are no y-intercepts and the function is symmetric with respect to the y-axis, sketch a U-shaped curve (parabola) that opens towards the positive x-axis, passing through the x-intercept and symmetric with respect to the y-axis.

Key Concepts

Intercepts in Graphing

When graphing a parabola, intercepts are crucial points of interest. The x-intercept is where the graph crosses the x-axis. To find it, set \(y = 0\) in the equation \(x = y^2 - 1\). Solving this gives \(x = -1\). Hence, the graph touches the x-axis at the point \((-1, 0)\).

The y-intercept is where the graph crosses the y-axis. For this, set \(x = 0\) and attempt to solve for \(y\). Given the equation \(0 = y^2 - 1\), solving for \(y\) results in no real solutions. This implies there are no y-intercepts for this equation. Identifying these intercepts simplifies plotting and understanding the graph's behavior.

Symmetry in Graphs

Symmetry is a useful property that makes graphing easier. A function symmetric concerning the y-axis means it mirrors itself on either side of this axis. To check symmetry for \(x = y^2 - 1\), replace \(y\) with \(-y\). This gives \(x = (-y)^2 - 1\), which simplifies back to \(x = y^2 - 1\).

Because the equation remains unchanged, it confirms the symmetry about the y-axis. This symmetry indicates that if you know the behavior of the graph on one side, you can replicate it on the other. For parabolas, this often provides a clearer picture of the whole graph by sketching just one half and reflecting it across the y-axis.

Sketching Graphs

Drawing a parabola involves understanding its intercepts and symmetry. Start by plotting the x-intercept. In this case, mark \((-1, 0)\) on the x-axis. Next, acknowledge the symmetry about the y-axis, which means the graph will be evenly distributed and mirror itself across this line.

Since the equation \(x = y^2 - 1\) doesn't possess any y-intercepts and is symmetric, you can visualize a U-shaped curve. This parabola opens horizontally, with its vertex at the x-intercept. Ensure the curve extends evenly on both sides of the y-axis. The understanding of intercepts and symmetry guides a more accurate and aesthetically precise sketch.