Problem 31 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find an equation of the line that passes through the given point and has the indicated slope. Sketch the line by hand. Use a graphing utility to verify your sketch, if possible. $$\left(-\frac{1}{2}, \frac{3}{2}\right), \quad m=0$$

Step-by-Step Solution

Answer

The equation of the line is \(y = 3/2\). It's a horizontal line that goes through the point \(-1/2, 3/2\).

Step 1: Apply the point-slope form

The point-slope form of a line's equation is given by \(y - y_1 = m(x - x_1)\). In this case, \(m = 0\), \(x_1 = -1/2\), and \(y_1 = 3/2\). When inserting these values into the formula, it simplifies to \(y = 3/2\), because any number multiplied by zero equals zero.

Step 2: Graph the line

The line \(y = 3/2\) runs horizontally at \(y = 3/2\) on the graph. Starting from the \(y\)-intercept that is 3/2, draw a horizontal straight line that runs left and right without ever changing its \(y\)-distance.

Step 3: Verify the graph

When possible, verify the graph with a graphing tool, which should also show an horizontal line at \(y = 3/2\).

Key Concepts

Point-Slope Form

Understanding the point-slope form of a line's equation can be a game-changer for graphing straight lines. It prominently features two pieces of key information – a point that the line passes through, and the line's slope. Formally, the equation is written as \( y - y_1 = m(x - x_1) \) where \( (x_1, y_1) \) represents the coordinates of the given point on the line, and \( m \) is the slope of the line.

This form is particularly useful because it allows you to plug in the coordinates of a known point and the slope value directly. If the slope (\b>m) is zero, as in our example, the equation simplifies marvellously. Multiplying zero by anything still gives you zero, which leads to the horizontal line equation \b>\( y = y_1 \) . To solidify your understanding, try plugging in other points and slopes into the point-slope form and see how it affects the equation.

Slope of a Line

The slope is a measure of how steep a line is and the direction it’s going. It’s simply the ratio of the vertical change to the horizontal change between two points on a line. Denoted by the letter \( m \) , it can be calculated by taking two points \b>\( (x_1, y_1) \) and \b>\( (x_2, y_2) \) on the line, and computing \( m = \frac{y_2 - y_1}{x_2 - x_1} \) .

In our exercise, the slope is given as zero. This means the line is not inclined at all—it's flat. A zero slope results in a horizontal line, and irrespective of the horizontal movement, the vertical height remains constant. If you are faced with a negative slope, the line will fall from left to right; a positive slope indicates a rise. Remember, a steeper line corresponds to a larger absolute slope value. Practicing with different slope values can help you predict the orientation of a line quickly!

Graphing Linear Equations

Graphing is a visual way to represent a linear equation and can reveal valuable insights into the nature of the line it describes. When graphing linear equations, we often use the y-intercept (where the line crosses the y-axis) and the slope. These components guide us in plotting the initial point and determining the line's direction and steepness.

For the linear equation we derived from the point-slope form, \( y = 3/2 \) , drawing the graph is straightforward. You simply locate the y-coordinate of 3/2 on the y-axis and draw a straight, horizontal line across the graph, perfectly parallel to the x-axis. This line doesn't tilt upwards or downwards because the slope is zero—it has no 'rise' over its 'run'. Always try to check your graph against a reliable graphing utility which can confirm whether your sketch accurately captures the essence of the linear equation.