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Chapter 1: Problem 22

Write an equation for the line with slope 5 and \(y\) -intercept -2

Short Answer

Expert verified
The equation of the line is \(y = 5x - 2\)

Step by step solution

01

Identify slope(m) and y-intercept(b)

From the problem, the slope \(m\) is given as 5 and the y-intercept \(b\) is given as -2
02

Substitute the slope and y-intercept into the formula

Substitute \(m=5\) and \(b=-2\) into the slope-intercept form of the equation \(y=mx + b\). This gives us \(y = 5x - 2\).

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a way to express linear equations. This format is useful because it directly shows the slope and y-intercept of the line, making it easy to graph and understand key characteristics of the line.
The general formula for the slope-intercept form is:
  • \( y = mx + b \)
Here, \( y \) represents the dependent variable, \( x \) is the independent variable, \( m \) is the slope of the line, and \( b \) is the y-intercept.
The slope-intercept form is advantageous for quickly identifying the line's steepness and where it crosses the y-axis.
Slope
The slope, often denoted by \( m \), indicates the line's steepness and direction. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line.
Mathematically, it is represented as:
  • \( m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} \)
A positive slope means the line rises from left to right, while a negative slope means it falls. A slope of zero indicates a horizontal line, and an undefined slope corresponds to a vertical line.
In our example, the slope is 5, suggesting a fairly steep line that ascends as it moves along the x-axis.
Y-Intercept
The y-intercept, represented by \( b \), is the point where the line crosses the y-axis. This value occurs when \( x = 0 \).
The importance of the y-intercept is its role in determining the starting point of the line on a graph.
In our exercise, the y-intercept is -2. This tells us that when \( x \) is 0, the value of \( y \) is -2, anchoring the line on the graph.
The y-intercept provides a crucial intersection point that helps in the graphing of lines and visualization of linear relationships.

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Most popular questions from this chapter

Use a graphing utility to draw several views of the graph of the function. Select the one that most accurately shows the important features of the graph. Give the domain and range of the function. $$f(x)=\left|x^{3}-3 x^{2}-24 x+4\right|$$

Give the domain and range of the function. $$f(x)=\sqrt{1+4 x^{2}}$$

Suppose that \(l_{1}\) and \(l_{2}\) are two nonvertical lines. If \(m_{1} m_{3}=\) \(-1,\) then \(l_{1}\) and \(l_{2}\) intersect at right angles. Show that if \(l_{1}\) and \(l_{2}\) do not interscet al right angles, then the angle \(\alpha\) between \(l_{1}\) and \(l_{2}\) (see Scction 1.4 ) is given by the formula $$\tan \alpha=\left|\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}\right|$$. HINT: Derive the identity $$\tan \left(\theta_{1}-\theta_{2}\right)=\frac{\tan \theta_{1}-\tan \theta_{2}}{1+\tan \theta_{1} \tan \theta_{2}}$$ by expressing the right side in terms of sines and cosines.

(a) Use a graphing utility to graph the polynomials $$\begin{aligned}&f(x)=x^{4}+2 x^{3}-5 x^{2}-3 x+1,\\\&g(x)=-x^{4}+x^{3}+4 x^{2}-3 x+2.\end{aligned}$$ (b) Based on your graphs in part (a), make a conjecture about the general shape of the graphs of polynomials of degree 4. (c) Test your conjecture by graphing $$f(x)=x^{4}-4 x^{2}+4 x+2 \text { and } g(x)=-x^{4}$$. Conjecture a property shared by the graphs of all polynomials of the form $$P(x)=x^{4}+a x^{3}+b x^{2}+c x+d$$. Make an analogous conjecture for polynomials of the form. $$Q(x)=-x^{4}+a x^{3}+b x^{2}+c x+d$$.

(a) Use a graphing utility to graph \(f_{A}(x)=A \cos x\) for several values of \(A ;\) use both positive and negative values. Compare your graphs with the graph of \(f(x)=\cos x\). (b) Now graph \(f_{B}(x)=\cos B x\) for several values of \(B\). since the cosine function is even, it is sufficient to use only positive values for \(B\). Use some values between 0 and 1 and some values greater than \(1 .\) Again, compare your graphs with the graph of \(f(x)=\cos x\). (c) Describe the effects that the coefficients \(A\) and \(B\) have on the graph of the cosine function.
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