Problem 11 - Precalculus

Question

Suppose (2,-3) is on the graph of \(y=f(x) .\) In Exercises \(1-18,\) use Theorem 1.7 to find a point on the graph of the given transformed function. $$ y=10-f(x) $$

Step-by-Step Solution

Answer

The point (2, 13) is on the graph of the transformed function y=10-f(x).

Step 1: Understand the Given Point

We are given that the point \((2, -3)\) is on the graph of \(y = f(x)\). This means when \(x = 2\), \(f(2) = -3\).

Step 2: Substitute the Given Point into Transformed Function

The transformed function is given as \(y = 10 - f(x)\). We substitute \(x = 2\) into this function to find the new \(y\)-value at this point.

Step 3: Calculate the New y-value

Using the original function information \(f(2) = -3\), substitute into the transformed function: \[ y = 10 - f(2) = 10 - (-3) = 10 + 3 = 13. \]

Step 4: Identify the New Point on the Transformed Function

The new point on the graph of the transformed function \(y = 10 - f(x)\) is \((2, 13)\).

Key Concepts

Graphing Functions

Graphing functions is a critical skill in mathematics, helping us visualize relationships between variables. When we create a graph of a function such as \(y = f(x)\), we plot points on a coordinate grid. Each point \((x, y)\) shows how the output \(y\) changes based on the input \(x\). For example, if the point \((2, -3)\) is on the graph, it tells us that when \(x\) is 2, \(f(x)\) is -3.

Understanding how to graph functions aids in comprehending transformations like shifts, stretches, and reflections. It also facilitates identifying key behaviors of functions, like intercepts and symmetries. Graphs bring to life the algebraic expressions that describe functions, making them an indispensable tool in both basic and advanced mathematics. Using them further allows us to interpret and predict the behavior of mathematical models efficiently.

Theorem 1.7

Theorem 1.7 is a valuable principle that deals with the transformations of functions. It states that if a point \((a, b)\) is on the graph of a function \(y = f(x)\), then the formula can be modified by transformations that involve adding, subtracting, multiplying, or dividing the function by a constant.

Take the example from the exercise: if we know \((2, -3)\) is on \(y = f(x)\), and we transform the function to \(y = 10 - f(x)\), the theorem assists us in finding the corresponding point on the new graph. Instead of solely relying on intuition, Theorem 1.7 ensures a systematic approach to finding new points after transformation, highlighting the logical and predictable nature of such modifications on functions.

Coordinates Transformation

In mathematics, coordinates transformation involves altering the existing coordinates of a graph due to certain operations performed on the function. This transformation changes how each point \((x, y)\) appears based on modifications made to \(f(x)\).

An example is given by transforming a function \(y = f(x)\) to \(y = 10 - f(x)\). Originally, we know the point \((2, -3)\) lies on \(f(x)\). By adjusting the function, we compute a new point \((2, 13)\) for \(y = 10 - f(x)\). This new point results from changing the original \(y\)-value \(f(2) = -3\) to \(10 - (-3) = 13\).

Such transformations are crucial for understanding function behavior and can involve scaling, translation, and reflection, which enables us to manipulate and adapt graphs accurately to represent different scenarios or data sets.

Function Evaluation

Function evaluation is the process of finding the output of a function for a given input. It's a fundamental concept that tells us what the function does to each specific input value. When we are told that \(f(2) = -3\), we are evaluating the function at \(x = 2\).

In our exercise, function evaluation helps us understand why \((2, -3)\) is on the graph of \(f(x)\). When we apply a transformation, such as changing \(y\) to \(10 - f(x)\), we need to evaluate this new function configuration at the same \(x\)-value. We substitute \(x = 2\) into the transformed function, calculating \(10 - (-3) = 13\).

Function evaluation allows us to accurately determine resultant points on a graph, ensuring that transformations produce valid and expected outcomes. It is vital for students to master this skill as it forms the basis of function work, preparing them for more complex mathematical concepts.