Problem 71 - A Graphical Approach to Precalculus with Limits

Question

Work each problem. If \(f(-2)=3,\) identify a point on the graph of \(f\)

Step-by-Step Solution

Answer

The point on the graph is \((-2, 3)\).

Step 1: Understand the Given Information

The problem states that when the input to the function \(f\) is \(-2\), the output is \(3\). This is represented by \(f(-2) = 3\).

Step 2: Identify the Point on the Graph

A point on the graph of a function \(f\) is of the form \((x, f(x))\). Since we are given \(f(-2) = 3\), the corresponding point is \((-2, 3)\).

Step 3: Write the Point

Using the information from the previous steps, write the point on the graph: \((-2, 3)\).

Key Concepts

Points on a Graph

A graph is a visual representation of a function in the coordinate plane. A point on this graph is given by an ordered pair, often in the form \((x, y)\), where \(x\) is the input (or independent variable), and \(y\) is the output (or dependent variable). Each point you see on the graph corresponds to one specific pair of input and output values of the function.

• The first number in the pair, \(x\), indicates the position on the horizontal axis (x-axis).
• The second number, \(f(x)\) or \(y\), indicates the position on the vertical axis (y-axis).

Understanding how to read and plot points on a graph is essential because it helps us visualize relationships expressed through a function. For instance, knowing \(f(-2) = 3\), we translate this into the point \((-2, 3)\) on the graph, where the graph crosses vertically at 3 when horizontally aligned with -2.

Function Evaluation

Function evaluation is the process of determining the output of a function for a specific input value. In simpler terms, it's like asking the function, "What do you give out when I give you this?" In the context of our example, we have the function \(f\) and are asked to evaluate it at \(-2\) which gives us \(f(-2) = 3\). Let's break this down:

• We identify the input, which in this instance is \(-2\).
• We use the function provided (in some cases, we would need to substitute \(-2\) into a given function formula).For this example, we're directly told the output.
• Finally, we capture the output value which here is \(3\). This output depends on the original formula or information given about \(f\).

Being adept at function evaluation enables you to fully engage with mathematical models and predictive calculations, as it forms the basis of interpreting function behavior.

Coordinate Geometry

Coordinate geometry, also referred to as analytic geometry, is the study of geometry using a coordinate system. This field combines algebra and geometry to analyze geometrical shapes and their properties using coordinates on a plane.
Key aspects of coordinate geometry include:

• Understanding the coordinate plane: This is a two-dimensional surface defined by the x and y axes.
• Recognizing points: Each point can be represented as \((x, y)\), where "x" indicates the horizontal position, and "y" indicates the vertical position of the point.
• Plotting points: By assigning coordinates like \((-2, 3)\), you can plot the precise location of a point on the graph, as we did with the function \(f(-2) = 3\).

Coordinate geometry provides the tools to graphically represent and solve problems involving distance, midpoint, slope, and more which are vital across various mathematical contexts, from basic graph drawing to predicting trends in data sets.