Problem 16 - Algebra and Trigonometry

Question

\(11-22\) a Determine an appropriate viewing rectangle for the equation, and use it to draw the graph. $$ y=\sqrt{12 x-17} $$

Step-by-Step Solution

Answer

Domain: \(x \geq \frac{17}{12}\); Range: \(y \geq 0\). Viewing rectangle: \((1, 10)\) for \(x\) and \((0, 10)\) for \(y\).

Step 1: Determine the domain of the function

The function given is a square root function, which exists for values of \(x\) where the expression inside the square root is non-negative. Thus, we need \(12x - 17 \geq 0\). Solve for \(x\):\[12x - 17 \geq 0 \12x \geq 17 \x \geq \frac{17}{12} \]This means that the domain of the function is \(x \geq \frac{17}{12}\).

Step 2: Determine the range of the function

Since the square root function outputs non-negative numbers, the range of this function is \(y \geq 0\).

Step 3: Choose an appropriate viewing rectangle

Considering the domain and range, we need \(x\) values starting from just below \(\frac{17}{12}\) (approximately 1.42) onwards, and \(y\) values starting from 0 upwards. A suitable viewing rectangle could be \(x\) from 1 to 10 and \(y\) from 0 to 10 ensuring both domain and range are captured.

Step 4: Sketch the graph within the viewing rectangle

Plot the graph within the chosen rectangle. Start from \(x = \frac{17}{12}\), where \(y = 0\), and draw the curve as \(x\) increases. The graph will rise smoothly starting at \(y = 0\). As \(x\) increases, the value under the square root increases, meaning \(y\) increases.

Key Concepts

Function Domain

In mathematics, a function's domain refers to the complete set of possible input values (often "x" values) that will result in meaningful output from the function. When working with a square root function like \(y = \sqrt{12x - 17}\), we must consider the values of \(x\) for which the expression inside the square root remains non-negative.

Importance of Domain:

• The values inside the square root must be greater than or equal to zero to produce real and valid numbers.
• The domain of \(y = \sqrt{12x - 17}\) requires that \(12x - 17 \geq 0\).

By solving this inequality, we find \(x \geq \frac{17}{12}\). This calculation indicates that for any \(x\) less than \(\frac{17}{12}\), the function does not exist as a real number since you would be attempting to take the square root of a negative number.

Function Range

The function range describes all possible output values (often "y" values) a function can produce. For the square root function \(y = \sqrt{12x - 17}\), the range helps us understand the extent of \(y\) values.

Identifying the Range:

• A square root function always sends out non-negative results because square root outputs are never negative.
• For our function, this means \(y \geq 0\) because the smallest value of \(y\) occurs when the expression under the square root is zero.

The range gives us insight into potential graph outputs, enabling better visualization and understanding of graph behavior.

Graphing Equations

Graphing equations visually represents a mathematical relationship, providing insights into specific solutions or the behavior of the function. For the equation \(y = \sqrt{12x - 17}\), effective graphing involves translating solutions from algebraic expressions to a meaningful visual format.

Steps to Graphing:

• Start plotting at \(x = \frac{17}{12}\), where \(y\) initially equals 0, because \(12x - 17\) becomes 0, resulting in a square root of 0.
• As the value of \(x\) increases, so does the value under the square root, making \(y\) increase and portray a rising curve.

Understanding the equation's behavior through graphing assists in conceptualizing how changes in \(x\) influence \(y\). It results in a curve that gives a physical depiction of function constraint by domain and range.

Viewing Rectangle

The concept of a viewing rectangle is crucial in graphing as it defines the portion of the graph visible on your graphing tool or screen. It is a frame capturing both domain and range, providing a complete picture of the function's behavior. When graphing \(y = \sqrt{12x - 17}\), selecting the appropriate viewing rectangle ensures proper representation of function characteristics.

Selecting a Good Viewing Rectangle:

• Choose an "x" range starting just below \(\frac{17}{12}\), since that's where our domain begins, to ensure we're seeing the whole picture.
• Given \(12x - 17\) yields \(0\) when \(x = 1.42\), setting "x" from 1 to 10 adequately captures it.
• For "y", starting from 0 upwards to 10 showcases the non-negative range.

By setting these ranges, graphing tools can better visualize function behaviors beginning from the valid points. This choice reflects both domain and range effectively within a graph's scope, aiding in comprehensive analysis.