91Ó°ÊÓ

The slope of a line is a crucial concept in understanding linear functions. It describes how steep the line is and is represented by the letter "m" in the linear equation format, which is generally written as \( f(x) = mx + b \). The slope essentially tells us how much the line rises (or falls) as it moves from left to right.

To calculate the slope, you need two distinct points on the line. Let's say these points are \((x_1, y_1)\) and \((x_2, y_2)\). The formula for the slope \(m\) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula finds the difference in the \(y\)-values divided by the difference in the \(x\)-values.

• Positive slope: The line rises as it moves from left to right.
• Negative slope: The line falls as it moves from left to right.
• Zero slope: The line is horizontal, with no rise or fall.

In this exercise, by using the points \((2, 5)\) and \((3, 7)\), we substitute into the slope formula: \( m = \frac{7 - 5}{3 - 2} = 2 \). This means that our line rises 2 units for every 1 unit it moves to the right.

The y-intercept of a line is where the line crosses the y-axis. It is an essential part of the linear function formula, as it determines the starting point of the line on the graph. When understanding any linear equation like \( f(x) = mx + b \), the y-intercept is represented by "b".

To find the y-intercept, you can choose any point on the line and substitute its coordinates into the linear equation when you already know the slope.
In this problem, once we calculated the slope \( m = 2 \), we used the point \((2, 5)\) to determine the y-intercept. Substituting into the equation:

• Take the point \((2, 5)\), so \( 5 = 2(2) + b \).
• Simplify: \( 5 = 4 + b \).
• Finally, solve for \(b\): \( b = 1 \).

That means the line crosses the y-axis at the point \((0, 1)\). This gives us the full linear function: \( f(x) = 2x + 1 \).

Function evaluation is about finding the output of a function for a given input. It's like asking what the function "does" when you put in a certain value for \(x\). In the context of linear functions, you apply the input to the formula to find the corresponding output.

Given the linear function \( f(x) = 2x + 1 \), you can evaluate the function for any value of \(x\) to find \(f(x)\). This is a straightforward process:

• Insert the value into the function wherever you see "\(x\)".
• Calculate the expression to find the result.

For the exercise, we needed to find \(f(4)\). Plugging \(4\) into the equation, we have:

Substitute \(x = 4\): \( f(4) = 2(4) + 1 \).

Simplify the expression: \( f(4) = 8 + 1 \).

Therefore, \( f(4) = 9 \).

Function evaluation allows you to determine specific points on your linear graph, verifying the line's behavior at different values of \(x\).