Problem 70 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The x-intercept of the equation is at (1,0), the y-intercept is at (0,1) and the graph has no symmetry with respect to x and y. The graph starts from point (1,0) and ascends as it moves leftwards, reaching its highest point at the y-intercept (0,1). The right half of the graph is undefined due to mathematical constraints.

Step 1: Identify Intercepts

To find the x-intercept, set \(y=0\) in the equation and solve for \(x\). By substituting \(y=0\), we get \(0 = \sqrt{1 - x}\). Squaring both sides, we have 0 = \(1 - x\), which simplifies to \(x = 1\). Thus the x-intercept is (1,0).\nTo find the y-intercept, set \(x=0\) in the equation and solve for \(y\). Substituting \(x=0\) we get \(y = \sqrt{1}\), which simplifies to \(y = 1\). Thus, the y-intercept is (0,1).

Step 2: Test for Symmetry

Check for symmetry by substituting \(-x\) for \(x\) and \(-y\) for \(y\). Thus, the equation becomes: \(-y = \sqrt{1 - (-x)}\). Squaring both sides results in \(y^2 = 1 + x\), which is not equivalent to the original equation, so the graph is not symmetric about the y-axis. Substitute \(x\) for \(-x\) and \(y\) for \(-y\) in the original equation, the equation becomes: \(-y = \sqrt{1 - x}\), which also doesn't match the original equation, so the graph is not symmetric about the x-axis. Hence, there is no symmetry with respect to x and y.

Step 3: Sketch the Graph

Having identified the intercepts and symmetry, one can now sketch the graph. The graph will start at point (1,0) on the x-axis and gradually ascend as \(x\) decreases until it reaches its highest point at the y-intercept, (0,1). The right side of the graph isn't defined for \(x > 1\) because you can't take the square root of a negative number, so there's only the left half of the graph.

Key Concepts

x-intercepts

X-intercepts are points where the graph touches or crosses the x-axis. At these points, the y-value is zero. To find the x-intercept, we set the equation equal to zero and solve for x. For the equation \(y=\sqrt{1-x}\), this means setting \(y = 0\):

• \(0 = \sqrt{1-x}\)
• Squaring both sides gives \(0 = 1-x\)
• Solve for \(x\) to find \(x = 1\)

Thus, the x-intercept is the point (1,0). This is the only point where the graph will meet the x-axis since the square root function only outputs non-negative numbers.

y-intercepts

Y-intercepts are where the graph hits the y-axis. At these points, the x-value is zero. To find the y-intercept in our equation \(y=\sqrt{1-x}\), we set \(x = 0\) and solve for \(y\):

• By substituting \(x=0\), we get \(y = \sqrt{1}\)
• This simplifies to \(y = 1\)

So, the y-intercept is at the point (0,1). This is the point on the y-axis where the square root function starts, indicating the highest point on this graph as it descends to the right.

symmetry testing

Symmetry testing helps us understand if the graph is mirrored across an axis or the origin. Let's explore symmetry for \(y=\sqrt{1-x}\):

• Y-axis symmetry: Replace \(x\) with \(-x\). If the equation remains unchanged, the graph is symmetric about the y-axis. For this equation, substituting gives: \(-y = \sqrt{1+x}\). Squaring both sides results in \(y^2 = 1 + x\), which does not equal the original equation. So, no y-axis symmetry.
• X-axis symmetry: Replace \(y\) with \(-y\). If the equation is the same, it is symmetric about the x-axis. Substituting results in \(-y = \sqrt{1-x}\), differing from the initial equation. Thus, no x-axis symmetry either.

The inability to achieve matching expressions indicates no symmetry around the x or y-axes. This informs us that the graph doesn’t mirror itself along either axis.

graph sketching

Now that we have identified the intercepts and determined symmetry, we can sketch the graph of \(y=\sqrt{1-x}\).

• The graph is defined only for \(x \leq 1\). You can't take the square root of a negative number, which keeps the function to the left of \(x=1\).
• Start plotting at x-intercept (1,0) and move left towards the y-intercept (0,1).
• Observe how it starts at \(x = 1\) at the x-axis and steadily rises to reach the y-intercept at \(y=1\) for \(x=0\).
• The graph resembles a sideways parabola opening leftward, characterized by its increasing nature as \(x\) moves from 1 to 0.

Sketching this helps visualize the gradual increase and the defined boundaries, providing clarity on how the graph should look and behave.