Problem 15 - A Graphical Approach to College Algebra

Question

Work each problem related to linear functions. (a) Evaluate \(f(-2)\) and \(f(4)\) (b) Graph \(f\). How can the graph of \(f\) be used to determine the zero of \(f ?\) (c) Find the zero of \(f\) $$f(x)=0.4 x+0.15$$

Step-by-Step Solution

Answer

(a) \(f(-2) = -0.65\), \(f(4) = 1.75\); (b) Line graph, zero is at intersection with x-axis; (c) Zero is \(x = -0.375\).

Step 1: Evaluate f(-2) and f(4)

To evaluate \(f(-2)\), substitute \(-2\) for \(x\) in the function \(f(x) = 0.4x + 0.15\):\[f(-2) = 0.4(-2) + 0.15 = -0.8 + 0.15 = -0.65.\]Similarly, to evaluate \(f(4)\), substitute \(4\) for \(x\):\[f(4) = 0.4(4) + 0.15 = 1.6 + 0.15 = 1.75.\]

Step 2: Sketching the Graph of f(x)

To graph \(f(x) = 0.4x + 0.15\), identify the y-intercept and the slope. The y-intercept is 0.15 (where \(x=0\)), and the slope is 0.4, meaning the line rises by 0.4 units for every 1 unit it moves to the right. Plot the y-intercept at \((0, 0.15)\) and use the slope to find another point, such as \((1, 0.55)\), then draw the line through these points.

Step 3: Determine Zero from Graph

The zero of a function, \(f(x)\), is the \(x\)-value where the graph intersects the x-axis (where \(f(x) = 0\)). By looking at the graph you sketched in Step 2, identify this point. The zero appears at the \(x\)-axis intersection.

Step 4: Finding the Zero Algebraically

Set \(f(x)\) equal to zero and solve for \(x\):\[0.4x + 0.15 = 0.\]Subtract 0.15 from both sides:\[0.4x = -0.15.\]Divide both sides by 0.4:\[x = \frac{-0.15}{0.4} = -0.375.\]

Key Concepts

Evaluating Functions

Evaluating a function for a given input means replacing the variable in the function equation with the specified number and calculating the result. For example, for the function \( f(x) = 0.4x + 0.15 \), evaluating \( f(-2) \) requires substituting \(-2\) for \(x\).

• Substitute \(-2\) into the function: \( f(-2) = 0.4(-2) + 0.15 \).
• Perform the multiplication: \( 0.4 \times -2 = -0.8 \).
• Add the constant: \(-0.8 + 0.15 = -0.65 \).

This provides the output of \(-0.65\) when \(x = -2\). Similarly, by substituting \(4\) for \(x\):

• \( f(4) = 0.4(4) + 0.15 \),
• \( 0.4 \times 4 = 1.6 \).
• Add the constant: \(1.6 + 0.15 = 1.75\).

The result is \(1.75\) for \(x = 4\). Evaluating functions helps us understand the output of a function at specific inputs, guiding us through understanding how the function behaves.

Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate plane and connecting them to form a straight line. The function \( f(x) = 0.4x + 0.15 \) is a linear equation because it forms a straight line when graphed.
To graph this function:

• Identify the y-intercept, which is the constant term \(0.15\). This is the point where the line crosses the y-axis, so plot the point at \((0, 0.15)\).
• The slope, \(0.4\), indicates that for every increase of 1 in \(x\), \(f(x)\) increases by 0.4. With the slope, start from \((0, 0.15)\), and move 1 unit right and 0.4 units up to locate another point, \((1, 0.55)\).
• Draw a line through the points \((0, 0.15)\) and \((1, 0.55)\). Extend the line across the graph.

This line represents all possible solutions for the function. Graphing helps in visually identifying function characteristics, such as slope and intercepts, and assessing the function's behavior over a range of values.

Finding Zeros of Functions

The zero of a function is the x-value where the function equals zero. In other words, it is where the graph of the function crosses the x-axis. Identifying this point is essential because it shows where the function has no influence or effect.
For \( f(x) = 0.4x + 0.15 \), to find the zero:

• Set the function equal to zero: \(0.4x + 0.15 = 0\).
• Subtract \(0.15\) from both sides to get \(0.4x = -0.15\).
• Divide by \(0.4\) to isolate \(x\): \(x = \frac{-0.15}{0.4} = -0.375\).

Thus, the zero of the function is \(-0.375\). On the graph of \(f(x)\), this zero is the point where the line crosses the x-axis. Finding zeros allows us to understand critical balancing points in a mathematical model, and they are vital in determining solutions to equations.