Problem 16 - Algebra and Trigonometry

Question

11–18 ? Find the x- and y-intercepts of the graph of the equation. $$ y=\sqrt{x+1} $$

Step-by-Step Solution

Answer

The y-intercept is (0, 1) and the x-intercept is (-1, 0).

Step 1: Find the y-intercept

To find the y-intercept, we set \( x = 0 \) and solve for \( y \). Substituting \( x = 0 \) into the equation \( y = \sqrt{x+1} \), we get:\[ y = \sqrt{0 + 1} = \sqrt{1} = 1 \].Thus, the y-intercept is \((0, 1)\).

Step 2: Find the x-intercept

To find the x-intercept, we set \( y = 0 \) and solve for \( x \). Substituting \( y = 0 \) into the equation \( 0 = \sqrt{x + 1} \), we square both sides to get:\[ 0^2 = (\sqrt{x+1})^2 \]\[ 0 = x + 1 \]By solving, we find:\[ x = -1 \].Thus, the x-intercept is \((-1, 0)\).

Key Concepts

Understanding the X-Intercept

The x-intercept of a graph is where the graph intersects or crosses the x-axis. At this point, the value of y is always zero. To find the x-intercept, we set the equation equal to zero and solve for x. This means: * Substitute 0 for y in the equation. * Solve for x. In our exercise, the equation given is \( y = \sqrt{x+1} \). When y is set to 0, the equation becomes \[0 = \sqrt{x+1} \]By squaring both sides, we eliminate the square root:\[0 = x+1 \]Solving for x, we subtract 1 from both sides to isolate x, resulting in \( x = -1 \). Therefore, the x-intercept, where the graph touches the x-axis, is at the point (-1, 0). This means when x is -1, y equals 0, confirming our understanding of the x-intercept.

Understanding the Y-Intercept

The y-intercept is the point where the graph intersects the y-axis. At this intersection, the value of x is always zero. To determine the y-intercept, replace x with zero in the equation and solve for y. In the problem at hand, the equation is \( y = \sqrt{x+1} \). By substituting x with 0, we have: \[y = \sqrt{0+1} \]This simplifies to \( y = \sqrt{1} \). Knowing that the square root of 1 is 1, we find \( y = 1 \). Thus, the y-intercept is the point (0, 1) on the graph. This tells us that when x is 0, y equals 1. It's the point where the graph meets the y-axis.

Graphing Equations with Intercepts

Graphing an equation involves plotting it on a coordinate plane. By understanding intercepts, we gain a crucial starting point for graphing any equation efficiently. Intercepts guide us to plot the curve accurately and help visualize the equation.To graph, follow these steps:

• Start by determining the intercepts. As seen, for this equation, they are the x-intercept at (-1, 0) and the y-intercept at (0, 1).
• Plot these intercepts on the coordinate plane.
• Recognize the nature of the function. The equation \( y = \sqrt{x+1} \) is a square root function which indicates a curve that starts from the x-intercept and extends to the first quadrant.
• Draw a smooth curve passing through the intercepts, noting the shape of the curve typical for square root functions.

Understanding intercepts gives a framework to predict the behavior of the graph beyond these points, ensuring an accurate representation.