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Chapter 4: Problem 2

Find an equation of the line containing the points \((1,4)\) and \((3,8)\)

Short Answer

Expert verified
The equation of the line is \(y = 2x + 2\)

Step by step solution

01

Identify the coordinates

The given points are \((1, 4)\) and \((3, 8)\). Let \(x_1, y_1\) be \(1, 4\) and \(x_2, y_2\) be \(3, 8\).
02

Calculate the slope

Use the formula for the slope \(m\) between two points: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given points, \[ m = \frac{8 - 4}{3 - 1} = \frac{4}{2} = 2 \]
03

Use the point-slope form

The point-slope form of the equation of a line is: \[ y - y_1 = m(x - x_1) \] Substituting \(m = 2\), \(x_1 = 1\), and \(y_1 = 4\), \[ y - 4 = 2(x - 1) \]
04

Simplify the equation

To write the equation in slope-intercept form \(y = mx + b\), distribute and solve for \(y\): \[ y - 4 = 2(x - 1) \] \[ y - 4 = 2x - 2 \] \[ y = 2x + 2 \]

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Key Concepts

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

Slope Formula
Understanding the slope formula is crucial in finding the equation of a line. The slope, often represented by the letter \( m \), measures the steepness or incline of a line.
The slope formula for two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
This formula gives the rise (change in \( y \)) over the run (change in \( x \)).
For example, if our points are \((1, 4)\) and \((3, 8)\), substituting into the formula, we get:
\[ m = \frac{8 - 4}{3 - 1} = \frac{4}{2} = 2 \]
So, the slope of our line is 2. This means for every unit increase in \( x \), \( y \) increases by 2 units.
Point-Slope Form
The point-slope form of a line equation is essential for creating an equation when you know one point on the line and the slope. The formula is:

\[ y - y_1 = m(x - x_1) \]
Here, \( (x_1, y_1) \) is a known point on the line, and \( m \) is the slope.
Using the slope from our earlier example (\( m = 2 \)) and the point \( (1, 4) \), we substitute these values into the formula:
\[ y - 4 = 2(x - 1) \]
This form is helpful because it immediately shows how the line behaves around the particular point \( (1,4) \).
Slope-Intercept Form
Converting the equation to slope-intercept form \( y = mx + b \) makes it easy to graph the line and understand its properties immediately.

To convert from point-slope form \( y - 4 = 2(x - 1) \):

First, distribute the slope 2 through the \( (x - 1) \):
\[ y - 4 = 2(x - 1) \]
\[ y - 4 = 2x - 2 \]

Next, solve for \( y \) by adding 4 to both sides:
\[ y = 2x - 2 + 4 \]
Finally, simplify the equation:
\[ y = 2x + 2 \]

In this form, it鈥檚 clear the line has a slope \( m = 2 \) and a y-intercept \( b = 2 \), meaning the line crosses the y-axis at \( (0,2) \).
Two Points on a Line
Finding the equation of a line requires understanding how to use two points on the line. Here鈥檚 how to use the points \( (1, 4) \) and \( (3, 8) \) step-by-step:

1. Identify the coordinates: Let \( x_1, y_1 = 1, 4 \) and \( x_2, y_2 = 3, 8 \).

2. Calculate the slope \( m \):
\[ m = \frac{8 - 4}{3 - 1} = 2 \]
3. Use the point-slope form with one of the points (we'll use \( (1, 4) \)):
\[ y - 4 = 2(x - 1) \]
4. Convert to slope-intercept form (if needed):
\[ y = 2x + 2 \]

By following these steps, you can always determine the line's equation passing through any two points.

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Most popular questions from this chapter

A projectile is fired from a cliff 200 feet above the water at an inclination of \(45^{\circ}\) to the horizontal, with a muzzle velocity of 50 feet per second. The height \(h\) of the projectile above the water is modeled by $$ h(x)=\frac{-32 x^{2}}{50^{2}}+x+200 $$ where \(x\) is the horizontal distance of the projectile from the face of the cliff. (a) At what horizontal distance from the face of the cliff is the height of the projectile a maximum? (b) Find the maximum height of the projectile. (c) At what horizontal distance from the face of the cliff will the projectile strike the water? (d) Graph the function \(h, 0 \leq x \leq 200\). (e) Use a graphing utility to verify the solutions found in parts (b) and (c). (f) When the height of the projectile is 100 feet above the water, how far is it from the cliff?

Why is the graph of a quadratic function concave up if \(a>0\) and concave down if \(a<0 ?\)

(a) Graph fand g on the same Cartesian plane. (b) Solve \(f(x)=g(x)\) (c) Use the result of part (b) to label the points of intersection of the graphs of fand \(g\). (d) Shade the region for which \(f(x)>g(x)\); that is, the region below fand above \(g\). \(f(x)=-2 x-1 ; \quad g(x)=x^{2}-9\)

53\. Simplify: \(\frac{5 x^{4}(2 x+7)^{4}-8 x^{5}(2 x+7)^{3}}{(2 x+7)^{8}}\)

Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value, and then find the value. \(f(x)=2 x^{2}+12 x-3\)
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