91Ó°ÊÓ

Understanding the slope of a linear function is crucial, as it describes how steep the line is. To calculate the slope, denoted as \( m \), you need two points on the line. In our exercise, we use the points (1, 2) and (3, 9). Mathematically, the slope is the 'rise over the run', or the change in \( y \) over the change in \( x \). The formula is \( m = \frac{(y_2-y_1)}{(x_2-x_1)} \).
By plugging in our values, we get \( m = \frac{(9-2)}{(3-1)} \), which simplifies to \( m = \frac{7}{2} \) or 3.5. This means for every one unit increase in \( x \), \( f(x) \) increases by 3.5 units.

Considering Other Points

In the textbook solution, we used specific points to calculate the slope, but it's important to know that we could use any two points from the table for the slope calculation. The slope should remain consistent because the relationship between \( x \) and \( f(x) \) is linear and constant.

The y-intercept is the point where the line crosses the y-axis. It is where \( x = 0 \). In a linear equation of the form \( f(x) = mx + b \), the y-intercept is represented by \( b \). After calculating the slope, we need to find the y-intercept to complete the equation of the line.
From our slope calculation, we know \( m = 3.5 \). We can use the point (1, 2) to find the y-intercept. We substitute these values into the equation to get \( 2 = 3.5(1) + b \), which can be rearranged to solve for \( b \), giving us \( b = -1.5 \). This value tells us that the line will cross the y-axis at -1.5.

Intercept Significance

The y-intercept is particularly important as it provides a starting value for the function when there is no horizontal movement, that is when \( x = 0 \). It's also useful for graphing the linear function, as you can start at the y-intercept and use the slope to find other points on the line.

Function evaluation is about finding the value of \( f(x) \) for a given input \( x \). With our linear function \( f(x) = mx + b \), once we know the values of \( m \) and \( b \), we can substitute any value for \( x \) to find the corresponding \( f(x) \).
In the exercise, after determining \( m = 3.5 \) and \( b = -1.5 \), we verify our findings by plugging another point from our table, such as \( x = 5 \). The calculation should be \( f(5) = 3.5*5 - 1.5 \), which gives us \( 16 \), aligning with the given table value.

Consistency Check

It's essential to check if the function is correct by using it to evaluate all points listed in the table. If the outputs match, we can be confident our slope and y-intercept are accurate. This step reinforces our understanding of the linear relationship by connecting the calculations with the graphical representation of the line.