Questions: (b) y = -x Make a table of values, and sketch the graph of the equation. Find the x - and y-intercepts. (If an answer does not exist, enter DNE.) x -intercept (x, y) = ( ) y -intercept (x, y) = ( ) Test for symmetry. (Select all that apply.) The graph is symmetric with respect to the x-axis. The graph is symmetric with respect to the y-axis. The graph is symmetric with respect to the origin. The graph is not symmetric with respect to the x-axis, the y-axis, or the origin.

(b) y = -x

Make a table of values, and sketch the graph of the equation.

Find the x - and y-intercepts. (If an answer does not exist, enter DNE.)
x -intercept (x, y) = ( )
y -intercept (x, y) = ( )

Test for symmetry. (Select all that apply.)
The graph is symmetric with respect to the x-axis.
The graph is symmetric with respect to the y-axis.
The graph is symmetric with respect to the origin.
The graph is not symmetric with respect to the x-axis, the y-axis, or the origin.

Transcript text: (b) $y=-|x|$ Make a table of values, and sketch the graph of the equation. Find the $x$ - and $y$-intercepts. (If an answer does not exist, enter DNE.) \[ \begin{array}{ll} x \text {-intercept } & (x, y)=(\square) \\ y \text {-intercept } & (x, y)=(\square) \end{array} \] Test for symmetry. (Select all that apply.) The graph is symmetric with respect to the $x$-axis. The graph is symmetric with respect to the $y$-axis. The graph is symmetric with respect to the origin. The graph is not symmetric with respect to the $x$-axis, the $y$-axis, or the origin.

Solution

Solution Steps

Step 1: Find the x-intercept

The x-intercept is the point where the graph crosses the x-axis, meaning y = 0. Solving the equation y = -|x| for y = 0 gives us 0 = -|x|, which simplifies to |x| = 0, and thus x = 0. Therefore, the x-intercept is (0, 0).

Step 2: Find the y-intercept

The y-intercept is where the graph crosses the y-axis, meaning x = 0. Substituting x = 0 into the equation y = -|x| gives us y = -|0|, which simplifies to y = 0. Therefore, the y-intercept is (0, 0).

Step 3: Test for symmetry

x-axis symmetry: If a graph is symmetric with respect to the x-axis, then if the point (x, y) is on the graph, so is (x, -y). Since y = -|x|, then replacing y with -y gives -y = -|x| or y = |x|. This is not the original equation, so the graph is not symmetric about the x-axis.
y-axis symmetry: If a graph is symmetric with respect to the y-axis, then if (x, y) is on the graph, so is (-x, y). Replacing x with -x gives y = -|-x| or y = -|x|. This is the same as the original equation, so it _is_ symmetric with respect to the y-axis.
Origin symmetry: If a graph is symmetric with respect to the origin, then if (x, y) is on the graph, so is (-x, -y). Replacing both x and y gives -y = -|-x| or -y = -|x|, simplifying to y = |x|. This is not the original equation, so the graph is not symmetric about the origin.