Problem 32 - Algebra and Trigonometry

Question

19–44 ? Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts and test for symmetry. $$ x+y^{2}=4 $$

Step-by-Step Solution

Answer

The x-intercept is (4,0); the y-intercepts are (0,2) and (0,-2). The graph is symmetric with respect to the x-axis.

Step 1: Choose Values for x

To make a table of values, choose several values for \(x\) and substitute them into the equation \(x + y^2 = 4\). For instance, consider \(x = 0, 1, 2, 3, 4\).

Step 2: Solve for y

For each chosen \(x\), solve the equation \(y^2 = 4 - x\) to find the corresponding \(y\) values. If \(y^2 = 4 - x\), then \(y = \pm \sqrt{4 - x}\). For example, if \(x = 0\), \(y = \pm 2\).

Step 3: Complete the Table of Values

Using the solutions from Step 2, complete the table with pairs of \((x, y)\) for each value of \(x\). For example, for \(x = 2\), \(y = \pm \sqrt{2}\), giving the points \((2, \sqrt{2})\) and \((2, -\sqrt{2})\).

Step 4: Sketch the Graph

Plot the points from the table on a coordinate grid and connect them smoothly to show the curve represented by the equation \(x + y^2 = 4\). This graph will resemble a sideways parabola opening to the left.

Step 5: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\): \(x = 4\). Therefore, the x-intercept is \((4, 0)\).

Step 6: Find the y-intercepts

To find the y-intercepts, set \(x = 0\) in the equation and solve for \(y\): \(y^2 = 4\). Thus, \(y = \pm 2\), giving the y-intercepts as \((0, 2)\) and \((0, -2)\).

Step 7: Test for Symmetry

For x-axis symmetry, substitute \(-(x)\) into the equation; not satisfied. For y-axis symmetry, substitute \(x, -y\); true for all points if \((-y)^2 = y^2\). This means the graph is symmetric with respect to the x-axis.

Key Concepts

Understanding x-intercepts in Graphs

The x-intercept of a graph is the point where the graph crosses the x-axis. At this intercept, the value of y is zero. To find the x-intercept of the equation \(x + y^2 = 4\), we set \(y = 0\) and solve for \(x\).
By substituting \(y = 0\) into \(x + y^2 = 4\), we obtain \(x = 4\). This reveals that the x-intercept of the graph is at the point \((4, 0)\).
It's important to remember that x-intercepts can help you understand where a graph touches or crosses the x-axis. This is crucial for analyzing and interpreting the behavior of a graph.

Exploring y-intercepts in Graphs

The y-intercept is where the graph crosses the y-axis, and at this point, the value of \(x\) is zero. To find the y-intercept for the equation \(x + y^2 = 4\), we set \(x = 0\) and solve for \(y\).
By substituting \(x = 0\) into the equation, we find \(y^2 = 4\), which gives us \(y = \pm 2\). This means there are two y-intercepts: \((0, 2)\) and \((0, -2)\).
Knowing the y-intercept(s) of a graph provides a starting point for sketching the graph and understanding part of its path through the coordinate plane.

The Role of Symmetry in Graphs

Symmetry in graphs indicates that one side is a mirror image of the other. Testing for symmetry can simplify graphing and analyzing an equation.
For the graph of \(x + y^2 = 4\), we test different forms of symmetry:

• For x-axis symmetry, replace \(y\) with \(-y\). If the equation remains unchanged, there's symmetry along the x-axis. Substituting \(-y\) into the equation confirms this, as it remains valid for all \(y\).
• For y-axis symmetry, substitute \(-x\) for \(x\), which does not work here.

This graph is symmetric with respect to the x-axis, which helps us predict the shape and features of the graph.

Creating a Table of Values

A table of values is an organized way to calculate and display points to graph a function or equation. This is particularly useful for sketching graphs.
For the equation \(x + y^2 = 4\), here's how you make a table of values:
1. Pick different values for \(x\), like 0, 1, 2, 3, and 4. Substitute each into the equation to determine \(y\).
2. Solve \(y^2 = 4 - x\) for each chosen \(x\). This gives positive and negative \(y\) values (if \(4 - x\) is non-negative).
Using particular values of \(x\), recording \((x, y)\) pairs helps visualize the function and prepare for graphing.

Solving Equations to Graph

Solving equations is a crucial part of graphing because it helps to find specific points that define the curve or line of a graph.
For \(x + y^2 = 4\), start by isolating terms for \(y\). Rearrange the equation to find \(y\) values for given \(x\) values. This way, you can identify the sets of points on the graph.
Solving this type of equation involves finding values for \(y\) that satisfy \(y^2 = 4 - x\), then determining \(y = \pm \sqrt{4 - x}\). With these solutions, each \(x\) yields two values of \(y\), showing the graph's mirrored nature and complexity. This procedure is key for accurately sketching the equation's graph and understanding the overall shape and relations between the variables.