Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 27-32: If the point \(P\) is on the graph of a function \(f\), find the corresponding point on the graph of the given function. $$ P(-2,1) ; \quad y=-3 f(2 x)-5 $$

Step-by-Step Solution

Answer

The corresponding point is \((-1, -8)\).

Step 1: Understanding the Original Point

The point \(P(-2, 1)\) is given on the graph of the function \(f(x)\). This means that \(-2\) is the input for the function \(f\), and the output is \(1\). In other words, \(f(-2) = 1\).

Step 2: Finding the Transformed Input

In the new function \(y = -3f(2x) - 5\), the input for the transformed function is \(2x\). We need to find the corresponding point for this function, which means finding the output when the function is evaluated at \(x = -2\). So first, solve \(2x = -2\) for \(x\). This gives \(x = -1\).

Step 3: Evaluating the Function f

We know from the original point \((x, f(x)) = (-2, 1)\) that \(f(-2) = 1\). Now, we need the input value for the new function: in terms of \(f\), the input is \(f(2x)\), so we need \(f(-2)\) because \(2 \times (-1) = -2\). Again, \(f(-2) = 1\).

Step 4: Applying the Transformation

Substitute the value \(f(2x) = 1\) into the new function \(y = -3f(2x) - 5\). Replacing \(f(2x)\) by \(1\), we get \[ y = -3(1) - 5 = -3 - 5 = -8. \]

Step 5: Identifying the New Point

Now that we know how the function transforms the original point into a new one, the \(x\) value remains \(-1\) (the input value), and the \(y\) value is \(-8\) as calculated. The corresponding point on the new graph is \((-1,-8)\).

Key Concepts

Graphing Functions

Understanding the graph of a transformed function involves visualizing how changes to the function's formula affect its shape and position in a coordinate plane. In this exercise, we start with the graph of a function \( f \) and a point \( P(-2, 1) \) already drawn on the graph.

When we look at the new function \( y = -3f(2x) - 5 \), it changes the original graph of \( f \). Such transformations can include reflections, stretches, compressions, and translations.

• Reflection: The negative sign \(-3\) causes a reflection over the x-axis, flipping the graph upside-down.
• Vertical Stretch: The coefficient \(3\) stretches the graph vertically, making peaks more pronounced.
• Horizontal Compression: Multiplying the input \(x\) by 2 compresses the graph horizontally, reducing its width.
• Vertical Translation: Subtracting 5 shifts the entire graph downward by five units.

By applying these transformations, we can predict how any given point on the original graph will appear on the new graph. The specific point \( (-2,1) \) changes to \( (-1,-8) \) after we follow these transformation rules.

Input-Output Mapping

The relationship between input and output in a function describes how each input value corresponds to a unique output value. For the function \( f \), the point \( (-2,1) \) indicates that when the input is \(-2\), the output is \(1\). This relationship can be altered through transformations, which adjust how inputs map to outputs.

In the transformed function \( y = -3f(2x) - 5 \), the input value processed by the function is altered. Instead of using \( x \) directly, we have \( 2x \) as the input. This means each input into the function is scaled by 2, creating a new input-output mapping.

With the equation \( 2x = -2 \), we find that the transformed input \( x \) becomes \(-1\). This input of \(-1\) maps to the output of the transformed function, which after computation, results in an output value of \(-8\). Thus, the point \( (-2, 1) \) on the original function graph maps to the point \( (-1, -8) \) on the new function graph.

Algebraic Manipulation

Algebraic manipulation is fundamental in analyzing how a function transforms, influencing both input values and resulting outputs. We begin with the expression for our transformed function \( y = -3f(2x) - 5 \), and by evaluating it step-by-step, we identify how each component reshapes the output.

The step-by-step process involves:

• Scaling Input: Transformation is initiated by scaling the input with \( 2x \), which needs separate consideration from the original point \(-2\).
• Solving for \(x\): By setting \(2x = -2\), we discover \(x = -1\), establishing the input's role in the transformed function.
• Substituting Values: After solving, we substitute back into the function; \(f(2x)\) equals \(f(-2)\) since \(2 imes -1 = -2\).
• Calculating Output: Finally, plug in \(f(2x) = 1\) to find \(y = -3(1) - 5 = -8\).

Through careful manipulation and substitution, we efficiently translate points from the original graph of \(f\) to the transformed function, helping in visualizing all algebraic impacts on the coordinate plane.