Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-20: Sketch the graph of the equation, and label the \(x\) - and \(y\)-intercepts. $$ y=\sqrt{x} $$

Step-by-Step Solution

Answer

The graph of \( y = \sqrt{x} \) passes through the origin, which is both the \( x \)- and \( y \)-intercept.

Step 1: Identify the Function

The given function is \( y = \sqrt{x} \). This function represents a square root curve, which means it will only be defined for \( x \geq 0 \). The graph of this function will be a curve starting from the origin and extending towards positive \( y \) values as \( x \) increases.

Step 2: Find the x-intercepts

To find the \( x \)-intercepts, set \( y = 0 \) and solve for \( x \). We have \( 0 = \sqrt{x} \). Squaring both sides, we find \( x = 0 \). So the \( x \)-intercept is at the point (0, 0).

Step 3: Find the y-intercepts

To find the \( y \)-intercepts, set \( x = 0 \) and solve for \( y \). We have \( y = \sqrt{0} \), which results in \( y = 0 \). Therefore, the \( y \)-intercept is also at (0, 0). The intercepts coincide at the origin.

Step 4: Sketch the Graph

Plot the point (0,0) on the graph. As \( x \) increases from 0, \( y = \sqrt{x} \) increases without bound, but more slowly than a linear function. Draw a curve starting at the origin and moving upwards to the right. This curve should not go below the x-axis or to the left of the y-axis, as \( x \) cannot be negative.

Step 5: Label the Intercepts

Highlight the intercept at (0, 0) on your sketch, labeling it as both the \( x \)-intercept and \( y \)-intercept, since they occur at the same point on this graph.

Key Concepts

Graphing

When graphing the function \( y = \sqrt{x} \), it's important to visualize how this square root function behaves. The curve begins at the origin point (0,0) and moves in an upward direction to the right. This is due to the fact that as \( x \) increases, \( y = \sqrt{x} \) increases as well, albeit at a decreasing rate. The nature of the square root function means it will grow slower compared to linear functions.

To effectively graph \( y = \sqrt{x} \):

• Start by plotting the point (0,0) which is the beginning of your curve.
• As \( x \) increases to positive values, continue to plot points such as (1,1), (4,2), and (9,3).
• Connect these points with a smooth curve that hugs the non-negative side of the x-axis, and also remains on the non-negative side of the y-axis.

X-intercept

The \( x \)-intercept of a graph is where the curve meets the x-axis, meaning the y-value at that point is zero. For the function \( y = \sqrt{x} \), to find the \( x \)-intercept, we set \( y \) equal to zero. Setting \( y = 0 \) gives us the equation \( 0 = \sqrt{x} \). Solving for \( x \), we square both sides resulting in \( x = 0 \). This is because the only non-negative number whose square root is zero is zero itself.

Therefore, the x-intercept for this function is located at the origin, (0,0).x The function starts from this intercept and extends only towards positive x-values.

Y-intercept

The \( y \)-intercept is where the graph of the function intersects the y-axis. This point occurs where \( x \) is zero. To find it for the function \( y = \sqrt{x} \), set \( x = 0 \) and solve for \( y \).Substituting in \( x = 0 \) gives us \( y = \sqrt{0} \), leading to \( y = 0 \). Hence, the y-intercept of \( y = \sqrt{x} \) is at the point (0,0).

In cases of the square root function, both the x-intercept and y-intercept are at the same point on the graph, emphasizing that the curve begins at the origin.

Functional Notation

Functional notation allows us to describe a function in a concise manner. For instance, \( y = \sqrt{x} \) uses functional notation to show a relationship between \( y \) and \( x \). In this particular function:

• The input, \( x \), represents real numbers that are equal to or greater than zero, since square roots of negative numbers aren't defined within the real number system.
• The output, \( y \), is the non-negative square root of \( x \), indicating how \( y \) changes as \( x \) varies.
• This notation is helpful for understanding how to transform or interpret the function graphically using different values of x to find corresponding values of y.

Remember, functional notation is key in clearly defining what mathematical operations are being performed on the variable \( x \) to obtain the resulting value of \( y \). It also helps when sketching the graph of the function and locating intercepts effectively.