Problem 8 Find the linear function \(f\) p... [FREE SOLUTION]

Chapter 2: Problem 8

Find the linear function \(f\) passing through the given points. (6,-12) and (-4,13)

Short Answer

Expert verified

The linear function is \(f(x) = -2.5x + 3\).

Step by step solution

Calculate the slope of the line

The first step in finding a linear function that passes through two given points is to calculate the slope () using the formula for the slope : \( = \frac{y_2 - y_1}{x_2 - x_1}\).Let's use the points (6, -12) and (-4, 13): \( = \frac{13 - (-12)}{-4 - 6} = \frac{25}{-10} = -2.5 \). The slope is -2.5.

Use the point-slope form of the equation

Now use the point-slope form of a line, which is \(y - y_1 = m(x - x_1)\). We can use either point, but let's use (6, -12).Substituting the values, we get: \( y - (-12) = -2.5(x - 6) \). Simplifying: \( y + 12 = -2.5(x - 6) \).

Simplify to slope-intercept form

Distribute and simplify the equation into slope-intercept form \(y = mx + b\):\( y + 12 = -2.5x + 15 \). Now, isolate \(y\) by subtracting 12 from both sides: \( y = -2.5x + 3 \). The linear function in the slope-intercept form is \(y = -2.5x + 3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope

In the context of a linear function, the slope is a crucial concept that tells us how steep the line is. It measures the rate of change along the line and is represented by the symbol "m". The formula to calculate the slope between two points, \( (x_1, y_1) \) and \( (x_2, y_2) \), is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Where:

\(y_2 - y_1\) represents the vertical change or rise
\(x_2 - x_1\) indicates the horizontal change or run

Consider the points (6, -12) and (-4, 13). Plugging these into the formula, we calculate:

The vertical change is \(13 - (-12) = 25\)
The horizontal change is \(-4 - 6 = -10\)

So, the slope is \[m = \frac{25}{-10} = -2.5\]This tells us that for every unit increase in x, y decreases by 2.5 units. A negative slope indicates a downward slant from left to right.

Point-Slope Form

The point-slope form is a handy tool when you know the slope and a point on the line. It is represented by the equation:\[y - y_1 = m(x - x_1)\]Here:

\(m\) is the slope
\((x_1, y_1)\) is a point through which the line passes

If we use the slope \(-2.5\) that we calculated, and one of the points, say (6, -12), we have:\[y - (-12) = -2.5(x - 6)\]This simplifies to:\[y + 12 = -2.5(x - 6)\]The point-slope form is useful for retaining the specific relationship between x and y through a known point, making adjustments easier before formulating the slope-intercept form.

Slope-Intercept Form

The slope-intercept form of a linear equation clearly shows the slope and the y-intercept. It is written as:\[y = mx + b\]In this form:

\(m\) is the slope of the line
\(b\) is the y-intercept, where the line crosses the y-axis

After using the point-slope form, we proceed to convert it into the slope-intercept form:Starting from \[y + 12 = -2.5(x - 6)\]Distribute the slope and simplify:\[y + 12 = -2.5x + 15\]To isolate y, subtract 12 from both sides:\[y = -2.5x + 3\]Now, the slope-intercept form reveals that the line has a slope of \(-2.5\) and crosses the y-axis at the point (0, 3). This format is particularly useful for quickly graphing a line or analyzing its behavior.

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91影视

Find the linear function \(f\) passing through the given points. (6,-12) and (-4,13)

Short Answer

Step by step solution

Calculate the slope of the line

Use the point-slope form of the equation

Simplify to slope-intercept form

Key Concepts

Slope

Point-Slope Form

Slope-Intercept Form

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