Problem 50 - Thinking Mathematically

Question

Evaluate \(f(x)\) for the given values of \(x\). Then use the ordered pairs \((x, f(x))\) from your table to graph the function. $$ \begin{aligned} &f(x)=x+1\\\ &\begin{array}{|r|l|} \hline {}{\underline{\phantom{xx}}}{\boldsymbol{x}} & \boldsymbol{f}(\boldsymbol{x})=\boldsymbol{x}+\mathbf{1} \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \\ \hline \end{array} \end{aligned} $$

Step-by-Step Solution

Answer

The ordered pairs are (-2,-1), (-1,0), (0,1), (1,2), (2,3). Plotting these points on a graph will yield a straight, upward sloping line.

Step 1: Calculating the Value for f(x) at x=-2

Substitute the value \(x = -2\) into the equation to find \(f(-2) = -2 +1 = -1.\) Hence, when \(x = -2\), \(f(x) = -1\).

Step 2: Calculating the Value for f(x) at x=-1

Substitute the value \(x = -1\) into the equation to find \(f(-1) = -1 + 1 = 0.\) Hence, when \(x = -1\), \(f(x) = 0.\)

Step 3: Calculating the Value for f(x) at x=0

Substitute the value \(x = 0\) into the equation to find \(f(0) = 0 + 1 = 1.\) Hence, when \(x = 0\), \(f(x) = 1.\)

Step 4: Calculating the Value for f(x) at x=1

Substitute the value \(x = 1\) into the equation to find \(f(1) = 1 + 1 = 2.\) Hence, when \(x = 1\), \(f(x) = 2.\)

Step 5: Calculating the Value for f(x) at x=2

Substitute the value \(x = 2\) into the equation to find \(f(2) = 2 + 1 = 3.\) Hence, when \(x = 2\), \(f(x) = 3.\)

Step 6: Plotting the Function

Now plot the function using the pairs of x-values and their corresponding calculated f(x)-values on a graph: (-2,-1), (-1,0), (0,1), (1,2), (2,3). You'll notice that these points form a straight, upward sloping line, which is characteristic of a linear function.

Key Concepts

Linear Equations

Linear equations form the backbone of algebra and are among the first encounters students have with algebraic concepts. In its simplest form, a linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be represented in the form of \(y = mx + b\), where \(m\) denotes the slope of the line, and \(b\) represents the y-intercept, which is the point where the line intersects the y-axis.

From this equation, we can glean valuable information about the graph of the line, such as its direction of increase or decrease, and where it sits on the grid. When graphing, we typically find two or more points that satisfy the equation and then draw a straight line through these points to represent the equation visually. Linear equations are used extensively across various fields like economics, physics, biology, and computer science, essentially anywhere that a straight-line relationship is a suitable approximation.

Function Evaluation

Function evaluation is an essential skill when working with algebraic expressions and equations. It involves substituting a given value of \(x\) into the function to find the corresponding value of \(y\), or \(f(x)\). For example, in the function \(f(x) = x + 1\), by substituting different values of \(x\), we get a set of outcomes that are the evaluated function values.This process is similar to following a recipe – \(x\) is your ingredient, and \(f(x)\) is the resulting dish. Every specific 'ingredient' you put in yields a unique 'dish' out. By systematically evaluating the function at various input values, a good understanding of the function's behavior can be developed. This knowledge is particularly useful when plotting its graph, as it provides us with a collection of points through which the graph will pass.

Ordered Pairs

What Are Ordered Pairs?

Ordered pairs are a fundamental part of graphing and coordinate geometry. An ordered pair \( (x, y) \) consists of two elements: the first component \(x\), which corresponds to the horizontal position on a graph (also known as the x-coordinate), and the second component \(y\), which denotes the vertical position (the y-coordinate).Ordered pairs are essentially coordinates that tell us the exact location of points on a Cartesian plane. When evaluating functions and plotting linear equations, we rely on these ordered pairs to transfer algebraic solutions to a visual graph. By knowing that the first number takes us along the x-axis, and the second number takes us up or down the y-axis, we can pinpoint any location on the graph with precision. These coordinates help us draw accurate graphs and understand the spatial relationships between different points.

Slope of a Line

Deciphering Slope

The slope of a line measures how steep the line is. Mathematically, it is calculated as the rise over the run, or the change in \(y\) over the change in \(x\). For the linear function \(f(x) = mx + b\), the slope is represented by \(m\). If the slope is positive, the line rises as one moves from left to right. Conversely, a negative slope indicates the line falls.

The concept of slope is pervasive in mathematics and is particularly important when dealing with linear functions. Knowing the slope of a line enables us to predict the behavior of the function. In addition, it allows us to compare the steepness of different lines and determine whether they are parallel or perpendicular. For instance, in the given example linear function \(f(x) = x + 1\), the slope is 1, which means for every unit increase in \(x\), \(f(x)\) increases by 1. This constant rate of change is characteristic of a straight line, which you can visually confirm on the graph.