Question

MATHEMATICAL CONNECTIONS Which equation has a graph that is a line passing through the point \((8,-5)\) and is perpendicular to the graph of \(y=-4 x+1\) ? (A) \(y=\frac{1}{4} x-5\) (B) \(y=-4 x+27\) (C) \(y=-\frac{1}{4} x-7\) (D) \(y=\frac{1}{4} x-7\)

Short answer

The equation that has a graph passing through \((8,-5)\) and is perpendicular to \(y=-4x+1\) is \(y=\frac{1}{4}x-7\).

Step-by-step solution

Determine the slope

The slope of the given line is -4, so the slope of any line perpendicular to it is the negative reciprocal of -4, i.e, \(\frac{1}{4}\).

Find the y-intercept

Substitute \((8,-5)\) into the slope-intercept form \( y=mx+c \), where m is \(\frac{1}{4}\). So, it's \(-5 = \frac{1}{4} * 8 + c\). Solving for c, we get \(c = -5 - \frac{1}{4} * 8 = -7\).

Write the Final Equation

With m as \(\frac{1}{4}\) and c as -7, our equation becomes \(y=\frac{1}{4}x-7\).

Key concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is an amazingly handy way to express straight lines. This format essentially tells us where our line will start on the y-axis and how steep the line is. It follows the formula: \(y = mx + c\). Here, \(m\) represents the slope, and \(c\) is the y-intercept, which is the point where the line touches the y-axis.

Imagine that slope, \(m\), as the tilt of the line. If you're climbing a hill, the slope tells you how steep it is. Positive \(m\) makes the line go upwards as you move left to right. Negative \(m\) makes the line go downwards. Zero slope (\(m = 0\)) represents a flat line, like a calm lake without waves.

Next is the y-intercept, \(c\), which tells us where the line will hit the y-axis. It's where you'll start if you're drawing the line on a graph. Understanding the slope-intercept form makes it so much easier to graph and analyze linear equations. When given a slope and a point, you can quickly find the y-intercept and write the equation of the line in this format."

Negative Reciprocal

The negative reciprocal is a fascinating concept when it comes to understanding perpendicular lines. When two lines are perpendicular, they intersect at right angles. A key property of perpendicular lines is that their slopes are negative reciprocals of each other.

To find the negative reciprocal of a number, you take the reciprocal (flip the fraction) and place a negative sign in front of it. For a whole number, you can imagine it over 1. For example, the slope -4 can be written as \(-\frac{4}{1}\), and its negative reciprocal is \(\frac{1}{4}\).

So, if you have a line with a slope of \(m\), a line perpendicular to it will have a slope of \(-\frac{1}{m}\). This relationship is quite satisfying in geometry and makes working with perpendicular lines more predictable and fun! Whenever you need to write the equation of a line that is perpendicular to another, knowing about negative reciprocals is essential for finding the correct slope.

Graphing Linear Equations

Graphing linear equations is like taking all the numbers and turning them into a picture. It falls into place beautifully with concepts like slope-intercept form. By plotting these equations on a graph, you unravel the visual relationships between variables.

Here's how it unfolds:

• Start by locating the y-intercept on the graph. This is the easy part since it’s where the line crosses the y-axis.
• Next, you use the slope to determine how to move from the y-intercept. From this point, use the slope to go upwards or downwards and either right or left, as determined by the sign and value of the slope.

Each step is like hopping from one solid point to another until the line gracefully stretches across the graph. Linear equations like \(y = \frac{1}{4}x - 7\) become more intuitive when you break them down on paper, providing a visual cue on how changes in parameters affect their graph. Graphing is not only a powerful way to understand math concepts but also a helpful skill for numerous practical applications in real life.