Problem 21 - Algebra and Trigonometry Real Mathematics, Real People

Question

The zero(s) of the function are given. Verify the zero(s) both algebraically and graphically. \(Function\) \(f(x)=4(3-x)\) \(Z e r o(s)\) \(x=3\)

Step-by-Step Solution

Answer

Yes, \(x=3\) is a zero of the function both algebraically and graphically.

Step 1: Algebraic Verification

Evaluate the function at \(x=3\). Substitute \(3\) into the function \(f(x)=4(3-x)\) to check if it equals zero. Doing so gives \(f(3)=4(3-3)=4(0)=0\). Thus, algebraically, \(x=3\) is a zero of the function, because substituting \(3\) into the function gives a result of zero.

Step 2: Graphical Verification

Draw the graph of the function \(f(x)=4(3-x)\). This is a linear function with a slope of -4 and a y-intercept of 12. The graph is a declining line that intersects the x-axis at \(x=3\). Therefore, graphically, one can see that \(3\) is indeed a zero of the function.

Key Concepts

Algebraic Verification

To verify the zeros of a function algebraically, we need to substitute the proposed zero into the function and see if the result is zero. For instance, consider the function \(f(x) = 4(3-x)\), and we want to verify if \(x=3\) is indeed a zero.

Here’s how you do it:

• Substitute \(x=3\) into the function: \(f(3) = 4(3-3)\).
• Simplify inside the parentheses: \(4(0)\).
• The result is \(0\), confirming that \(x=3\) is a zero of the function.

In algebraic terms, a zero is a solution to the equation \(f(x) = 0\). When \(f(3) = 0\), it confirms that \(x=3\) balances the equation, proving it is indeed a zero. This method is definitive because it directly calculates the outcome using algebraic principles.

Graphical Verification

Graphical verification involves visually inspecting the graph of the function to confirm the location of its zeros. For the function \(f(x) = 4(3-x)\), which is linear, you can graph it easily.

These are the steps you can follow:

• Understand the basic form of the linear equation, which suggests a straight line: \(y = mx + b\).
• Your function can be rewritten in the form \(f(x) = -4x + 12\).
• Plot the y-intercept at 12 on the y-axis and use the slope \(-4\) to determine the line's angle.
• Draw the line which declines downwards, crossing the x-axis at \(x=3\).

By plotting this, you can visually confirm that the graph intersects the x-axis at \(x=3\). This intersection point on the x-axis indicates the zero of the function. Visual representation through graphing enhances understanding by providing a clear and intuitive insight of where the function equals zero.

Linear Function Analysis

To better grasp linear functions like \(f(x) = 4(3-x)\), it's essential to understand their general characteristics and components. Linear functions produce straight lines when graphed and have a constant rate of change.

Key features of linear functions include:

• Slope: This is the rate at which the function increases or decreases. In our function, the slope is -4. It tells us that for every 1 unit increase in \(x\), \(f(x)\) decreases by 4 units.
• Intercept: The y-intercept is a point where the line crosses the y-axis, which is at (0,12) for our function. It represents the value of \(f(x)\) when \(x=0\).
• Zeros: These are the x-values where the function equals zero. For \(f(x) = 4(3-x)\), \(x=3\) is the zero.

Understanding these elements allows for quick and effective analysis of linear functions. It helps to predict the graph's shape and behavior. By examining the slope and intercept, one can construct the graph and identify important points like zeros without needing detailed calculations every time.