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Chapter 2: Problem 6

Graphing Functions Sketch a graph of the function by first making a table of values. $$f(x)=4-2 x$$

Short Answer

Expert verified
Plot points for selected x-values, draw a line through them to sketch the linear graph.

Step by step solution

01

Understand the function

The function given is a linear function in the form \( f(x) = 4 - 2x \). This means it has a slope of -2 and a y-intercept of 4.
02

Choose values for x

Select a range of x-values to substitute into the function. For example, choose \(x = -1, 0, 1, 2, 3\) to get a clear picture of the graph.
03

Create the table of values

Calculate the corresponding y-values for each chosen x-value:- If \(x = -1\), then \( f(-1) = 4 - 2(-1) = 6 \).- If \(x = 0\), then \( f(0) = 4 - 2(0) = 4 \).- If \(x = 1\), then \( f(1) = 4 - 2(1) = 2 \).- If \(x = 2\), then \( f(2) = 4 - 2(2) = 0 \).- If \(x = 3\), then \( f(3) = 4 - 2(3) = -2 \).
04

Plot the points on a graph

Using the table of values, plot the points \((-1, 6), (0, 4), (1, 2), (2, 0), (3, -2)\) on a coordinate plane.
05

Draw the graph

Connect the plotted points with a straight line, extending it in both directions, since linear functions have infinite length in both directions. Ensure the line reflects the slope of -2.

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Key Concepts

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

linear equations
A linear equation is like a straight highway; it goes in one constant direction. In math, a linear equation refers to an equation that creates a straight line when graphed. The standard form of a linear equation is often given as \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept.

Here are some keys to understanding linear equations:
  • They have no exponents higher than one. This means terms like \(x^2\) or \(x^3\) won't appear in a linear equation.
  • When graphed, they will always produce a straight line.
  • The relationship between the variables is proportional, which is why the equation gives a constant slope.
Understanding linear equations is important because they are found everywhere in real life, from calculating speeds to budgeting finances and more.
slope-intercept form
The slope-intercept form is a way of writing linear equations, which makes it easy to identify important features of the line quickly. It's written as \( y = mx + b \).

The components of this form are:
  • \(m\) is the slope of the line, showing the steepness and direction of the line. In the equation \( y = -2x + 4 \), the slope is \(-2\). A negative slope means the line goes downwards from left to right.
  • \(b\) is the y-intercept, which is where the line crosses the y-axis. In our example, the y-intercept is \(4\), meaning the line hits the y-axis at the point \((0, 4)\).
The slope-intercept form is very handy for graphing because it gives you both the starting point on the y-axis and the direction to move to plot the line.
coordinate plane
The coordinate plane is a two-dimensional surface on which points are plotted. It鈥檚 like a large map divided into four quadrants by the x-axis (horizontal line) and the y-axis (vertical line). Understanding how to use this plane is essential for graphing linear functions.

Key features to know about the coordinate plane:
  • The point where the x-axis and y-axis intersect is called the origin, located at \((0, 0)\).
  • Any point on the coordinate plane is described by an ordered pair \((x, y)\).
  • When you plot a function, you place each point by moving 'x' units horizontally and 'y' units vertically, starting from the origin.
Utilizing the coordinate plane effectively helps in visualizing the behavior and shape of the linear equations.
table of values
A table of values is a simple way to generate points that can be plotted on the coordinate plane to visualize a linear equation. It involves choosing x-values, calculating their corresponding y-values, and writing them down as coordinate pairs.

Creating a table of values:
  • Choose some x-values, either randomly or within a specific range.
  • Substitute each x-value into the linear equation to find the y-value.
  • Record the results as ordered pairs \((x, y)\).
For instance, using the function \(f(x) = 4 - 2x\), if we pick x-values like -1, 0, 1, 2, and 3, we can calculate the corresponding y-values to create coordinate pairs: \((-1, 6)\), \((0, 4)\), \((1, 2)\), \((2, 0)\), \((3, -2)\). This table forms the blueprint for drawing the graph on the coordinate plane.

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Most popular questions from this chapter

A savings account earns \(5 \%\) interest compounded annually. If you invest \(x\) dollars in such an account, then the amount \(A(x)\) of the investment after one year is the initial investment plus \(5 \% ;\) that is, $$A(x)=x+0.05 x=1.05 x$$ Find $$\begin{aligned} &A \circ A\\\&A \circ A \circ A\\\&A \circ A \circ A \circ A\end{aligned}$$ What do these compositions represent? Find a formula for what you get when you compose \(n\) copies of \(A\).

Graph the function in each of the given viewing rectangles, and select the one that produces the most appropriate graph of the function. \(g(x)=x^{2}-x-20\) (a) \([-2,2]\) by \([-5,5]\) (b) \([-10,10]\) by \([-10,10]\) (c) \([-7,7]\) by \([-25,20]\) (d) \([-10,10]\) by \([-100,100]\)

Graphing Functions Sketch a graph of the function by first making a table of values. $$f(x)=1+\sqrt{x}$$

Graphing Functions Sketch a graph of the function by first making a table of values. $$f(x)=\sqrt{x-2}$$

As blood moves through a vein or an artery, its velocity \(v\) is greatest along the central axis and decreases as the distance \(r\) from the central axis increases (see the figure). The formula that gives \(v\) as a function of \(r\) is called the law of laminar flow. For an artery with radius \(0.5 \mathrm{cm},\) the relationship between \(v\) (in \(\mathrm{cm} / \mathrm{s}\) ) and \(r\) (in \(\mathrm{cm}\) ) is given by the function $$v(r)=18,500\left(0.25-r^{2}\right) \quad 0 \leq r \leq 0.5$$ (a) Find \(v(0.1)\) and \(v(0.4)\) (b) What do your answers to part (a) tell you about the flow of blood in this artery? (c) Make a table of values of \(v(r)\) for \(r=0,0.1,0.2,0.3\) \(0.4,0.5\) (d) Find the net change in the velocity \(v\) as \(r\) changes from \(0.1 \mathrm{cm}\) to \(0.5 \mathrm{cm}\)
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