Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-20: Sketch the graph of the equation, and label the \(x\) - and \(y\)-intercepts. $$ y=\sqrt{x}-4 $$

Step-by-Step Solution

Answer

The graph starts at \((0, -4)\) (y-intercept) and passes through \((16, 0)\) (x-intercept).

Step 1: Identify the Type of Function

The given equation is \(y = \sqrt{x} - 4\). Recognize that this is a square root function, which typically has a graph that starts at a certain point and then increases gradually.

Step 2: Determine the Domain of the Function

The expression under the square root, \(\sqrt{x}\), must be non-negative, which means \(x \geq 0\). Therefore, the domain of this function is \([0, \infty)\).

Step 3: Find the Y-Intercept

To find the \(y\)-intercept, substitute \(x = 0\) into the equation: \(y = \sqrt{0} - 4 = -4\). The \(y\)-intercept is at \((0, -4)\).

Step 4: Find the X-Intercept

To find the \(x\)-intercept, set \(y = 0\) and solve for \(x\): \(0 = \sqrt{x} - 4\). Solve \(\sqrt{x} = 4\) by squaring both sides: \(x = 16\). Thus, the \(x\)-intercept is at \((16, 0)\).

Step 5: Sketch the Graph

Plot the \(y\)-intercept \((0, -4)\) and the \(x\)-intercept \((16, 0)\) on a graph. The curve starts at \((0, -4)\) and rises gradually towards the right. The shape resembles the right half of a sideways parabola opening to the right.

Step 6: Label the Intercepts

Label the \(y\)-intercept as \((0, -4)\) and the \(x\)-intercept as \((16, 0)\) on your sketch for clarity.

Key Concepts

Understanding X-Intercepts

The x-intercepts of a graph are the points where the curve crosses the x-axis. To find the x-intercept for a square root function like \(y = \sqrt{x} - 4\), set \(y = 0\) and solve for \(x\). This gives us the equation \(0 = \sqrt{x} - 4\).

• Add 4 to both sides to get \(\sqrt{x} = 4\).
• Square both sides to eliminate the square root, resulting in \(x = 16\).

Therefore, the x-intercept is \((16, 0)\). This means that, at this x-value, the graph touches the x-axis. It's essential to understand that x-intercepts are crucial for determining where the graph crosses or touches the x-axis, reflecting points where the output value \(y\) is zero.

Identifying Y-Intercepts

The y-intercept is the point where the graph crosses the y-axis. This point occurs at \(x = 0\). For the function \(y = \sqrt{x} - 4\), substitute \(x = 0\) to find the y-intercept.

• The equation simplifies to \(y = \sqrt{0} - 4 = -4\).

Thus, the y-intercept is \((0, -4)\). This indicates that when no value is input for \(x\), the function produces an output of -4. It's crucial to determine the y-intercept because it tells you the starting value of the function—essentially, where the graph will begin on the y-axis.

Defining the Function Domain

The domain of a function consists of all possible input values, or x-values, that the function can accept without resulting in an undefined or imaginary output. For square root functions like \(y = \sqrt{x} - 4\), the expression inside the square root, \(x\) in this case, must be non-negative because square roots of negative numbers are not defined in the set of real numbers.

• Set the expression inside the square root, \(x \geq 0\), because it cannot be negative.

Accordingly, the domain of \(y = \sqrt{x} - 4\) is \([0, \infty)\). This tells you that the graph starts at \(x = 0\) and extends indefinitely to the right along the x-axis. Understanding the domain is essential in graphing, as it dictates the extent of the x-values that the function will cover.

Graph Sketching Made Simple

Graph sketching is the process of visualizing a function on the coordinate plane. For the function \(y = \sqrt{x} - 4\), follow these straightforward steps: 1. **Plot Intercepts**: Start by plotting the intercepts on the graph. Mark the y-intercept at \((0, -4)\) and the x-intercept at \((16, 0)\). 2. **Draw The Curve**: Understanding that \(y = \sqrt{x}\) typically starts at the point \((0,0)\) and gradually increases in a curved fashion, recognize that \(y = \sqrt{x} - 4\) will shift this curve downwards by 4 units. Hence, it begins at \((0, -4)\) and curves upwards. 3. **Smooth Curve**: Bring out a smooth, gradual curve from the y-intercept towards the x-intercept, mimicking the gradual rise typical of square root functions. 4. **Labelling**: Ensure all points, especially the intercepts, are clearly labeled for clarity.Grasping graph sketching helps you to see how functions behave visually, and practicing these steps will make you confident in translating the mathematical expression of a function onto a graph.