Problem 46 - Algebra and Trigonometry

Question

\(41-48\) Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$ y=\sqrt{x+1} $$

Step-by-Step Solution

Answer

x-intercept: (-1,0), y-intercept: (0,1).

Step 1: Finding the y-intercept

To find the y-intercept of the equation, we need to set \( x = 0 \) in the equation and solve for \( y \). Substitute \( x = 0 \) into \( y = \sqrt{x + 1} \) to get \( y = \sqrt{0 + 1} = \sqrt{1} = 1 \). Therefore, the y-intercept is at the point \((0, 1)\).

Step 2: Finding the x-intercept

To find the x-intercept of the equation, we need to set \( y = 0 \) and solve for \( x \). Set \( y = 0 \) in the equation \( y = \sqrt{x + 1} \) which gives us \( 0 = \sqrt{x + 1} \). This implies that \( x + 1 = 0 \). Solve for \( x \) to get \( x = -1 \). Therefore, the x-intercept is at the point \((-1, 0)\).

Key Concepts

Understanding the X-Intercept

The x-intercept of a graph is a key point where the graph crosses the x-axis. This happens when the value of y is zero. To find where this occurs on the graph of an equation, we set y to zero and then solve the equation for x. For the given equation, \( y = \sqrt{x + 1} \), by setting \( y = 0 \), it transforms to \( 0 = \sqrt{x + 1} \). Simplifying, we find \( x + 1 = 0 \) leads to \( x = -1 \). Thus, the x-intercept for this function is at the point \((-1, 0)\). This point and the process of finding it are important because they show where the function touches the horizontal axis. It is a crucial part of understanding the behavior of a graph as it reveals one of the boundaries where the graph changes from real positive or negative values to zero.

Exploring the Y-Intercept

The y-intercept of a graph is the point where the graph crosses the y-axis. At this particular point, the value of x is always zero. Finding the y-intercept involves substituting \( x = 0 \) into the equation and solving for y. For the equation \( y = \sqrt{x + 1} \), placing \( x = 0 \) gives \( y = \sqrt{0 + 1} \), which simplifies to \( y = 1 \). Thus, the y-intercept of this function occurs at the point \((0, 1)\). This process highlights how intercepts provide insights into the initial state of a function as it begins to plot on the graph, setting a baseline point of reference from which to explore further changes.

Insights into the Square Root Function

Square root functions are unique because they involve radicals and are only defined for values where what's inside the root is non-negative (greater than or equal to zero). For the function \( y = \sqrt{x + 1} \), this means that \( x + 1 \geq 0 \), leading to the domain of \( x \geq -1 \). This specific aspect of its domain is critically important because it restricts the x-values you can use when plotting the graph.
Characteristics of square root functions include:

• They always start at a defined point on the x-axis and increase thereafter.
• They are slow to rise initially, accelerating as x increases.
• The graph is typically curved and approaches infinity as x grows larger.

Understanding these properties helps in visualising and sketching the graph without plotting every possible point.