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Chapter 1: Problem 52

Identify any intercepts and test for symmetry. Then sketch the graph of the equation. $$y=\sqrt{1-x}$$

Short Answer

Expert verified
The intercepts are at x = 1 (X-intercept) and y = 1 (Y-intercept). The graph is not symmetric. The graph starts at (1,0) and extends to the left, getting closer to the y-axis but never crossing it.

Step by step solution

01

Finding the X-intercept

The x-intercept is found by setting y to 0 in the given function and solving for x. So, setting \( y = 0 \), we get : \( 0 = \sqrt{1 - x} \). Squaring both sides, we have \( 1 - x = 0 \). Solving this, we get \( x = 1 \). So, the x-intercept is \( x = 1 \).
02

Finding the Y-intercept

The y-intercept is found by setting x to 0 in the function and solving for y. Substituting \( x = 0 \) into the equation, we get \( y = \sqrt{1 - 0} \). So the y-intercept is \( y = 1 \).
03

Checking for Symmetry

To test for symmetry, replace \( x \) with \( -x \) in the equation and simplify. After substitution, you get \( y = \sqrt{1 - (-x)}\), which simplifies to \( y = \sqrt{1 + x} \). This is not the same as the original function, and as such, it does not have symmetry.
04

Sketching the Graph

Plot the intercepts from Step 1 and Step 2 on a graph, and remember that the square root function is reflected across the y-axis and shifted 1 unit to the right. The graph starts at (1,0) and extends to the left, getting closer to the y-axis but never crossing it.

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