Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 15-16: Sketch the graph of the line through \(P\) for each value of \(m\). $$ P(-2,4) ; \quad m=1,-2,-\frac{1}{2} $$

Step-by-Step Solution

Answer

Three lines with equations: \(y = x + 6\), \(y = -2x\), and \(y = -\frac{1}{2}x + 3\) through point \(P(-2, 4)\).

Step 1: Identify the equation of a line

To sketch a line, we need an equation. For a line passing through a point \(P(x_1, y_1)\) with slope \(m\), the equation is given by the point-slope form \(y - y_1 = m(x - x_1)\).

Step 2: Substitute point P(-2, 4) into the equation

Substitute \(x_1 = -2\) and \(y_1 = 4\) into the point-slope form. The equation becomes \(y - 4 = m(x + 2)\).

Step 3: Solve for y with m=1

Substitute \(m = 1\) to find the equation of the first line. This gives us \(y - 4 = 1(x + 2)\). Solving this yields \(y = x + 6\).

Step 4: Sketch the line for m=1

Plot the line \(y = x + 6\). This line will pass through point \(P(-2, 4)\) and have a slope of \(1\), indicating it rises 1 unit for every unit it runs to the right.

Step 5: Solve for y with m=-2

Substitute \(m = -2\) to find the equation of the second line. This gives us \(y - 4 = -2(x + 2)\). Solving this yields \(y = -2x\).

Step 6: Sketch the line for m=-2

Plot the line \(y = -2x\). This line will pass through \(P(-2, 4)\) and have a slope of \(-2\), indicating it falls 2 units for every unit it runs to the right.

Step 7: Solve for y with m=-\frac{1}{2}

Substitute \(m = -\frac{1}{2}\) to find the equation of the third line. This gives us \(y - 4 = -\frac{1}{2}(x + 2)\). Solving this yields \(y = -\frac{1}{2}x + 3\).

Step 8: Sketch the line for m=-\frac{1}{2}

Plot the line \(y = -\frac{1}{2}x + 3\). This line will pass through \(P(-2, 4)\) and have a slope of \(-\frac{1}{2}\), indicating it falls 0.5 units for every unit it runs to the right.

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is a handy way to represent a line when you know the slope and a point on the line. This form is given by:\[ y - y_1 = m(x - x_1) \] where

• \((x_1, y_1)\) is a point on the line
• \(m\) is the slope of the line

To use this form, you substitute the coordinates of a point (in this case, point \(P(-2, 4)\)) and a known slope \(m\) into the equation. This method is beneficial because it directly integrates the slant (or steepness) of the line along with a specific location through which the line passes. Once the equation is set up, you can easily convert it to another form, like the slope-intercept form, to graph the line visually.

Slope-Intercept Form

The slope-intercept form is perhaps the most straightforward form to work with when it comes to graphing linear equations. It is expressed as:\[ y = mx + b \] where

• \(m\) represents the slope of the line
• \(b\) provides the y-intercept, which is the point where the line crosses the y-axis

By converting the point-slope form equation to the slope-intercept form, it's easier to sketch the graph. This transformation makes it clear how steep the line is and at which point it touches the y-axis. For example, when solving with a slope \(m=1\), we find our equation becomes \(y = x + 6\). The line confidently says, "I rise 1 step upward for every step I take right, and I start my journey at the number 6 on the y-axis." This insight helps in quickly drawing the graph on a coordinate plane.

Slope of a Line

Understanding the slope of a line is crucial in mastering linear equations. The slope \(m\) tells you how steep the line is and in which direction it slants. It is calculated as the ratio of the vertical change, or "rise," to the horizontal change, or "run," between two points on a line:\[ m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} \] Here's how you can visualize it:

• A positive slope means the line is ascending as it moves from left to right.
• A negative slope indicates descending in the same direction.
• A zero slope reflects a horizontal line.
• An undefined slope shows a vertical line.

In the given exercise, different values of \(m\) show how the line's angle changes while still crossing through \(P(-2, 4)\). With \(m=1\), the line goes uphill at a 45° angle. With \(m=-2\), it swoops down more steeply. And with \(m=-\frac{1}{2}\), the slope is gentler in its descent. Knowing the slope lets you predict and draw the line behavior on a graph easily.