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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.

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 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.