Problem 55 Write a formula for \(f(x)=x^{2}... [FREE SOLUTION]

91影视

Precalculus: An Investigation of Functions

David Lippman, Melonie Rasmussen

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 1: Problem 55

Write a formula for $f(x)=x^{2}$ horizontally compressed by a factor of $\frac{1}{2},$ then shifted to the right 5 units and up 1 unit.

Short Answer

Expert verified

The formula is $ f(x) = 4(x-5)^2 + 1 $.

Step by step solution

Start with the Basic Function

The basic function we start with is \[ f(x) = x^2. \] This is the standard quadratic function.

Apply Horizontal Compression

To compress the function horizontally by a factor of $ \frac{1}{2} $, we replace $ x $ in \[ f(x) = x^2 \] with $ 2x $. Thus, the function becomes \[ f(x) = (2x)^2 = 4x^2. \] This compresses the graph towards the y-axis.

Shift Right by 5 Units

To shift the function 5 units to the right, we replace $ x $ in \[ f(x) = 4x^2 \]with $ x - 5 $. This gives the new function\[ f(x) = 4(x - 5)^2. \] The graph moves right by 5 units along the x-axis.

Shift Up by 1 Unit

To shift the function up by 1 unit, we add 1 to \[ f(x) = 4(x - 5)^2. \] Thus, the final function is \[ f(x) = 4(x - 5)^2 + 1. \] The entire graph moves upwards by 1 unit on the y-axis.

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Key Concepts

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

Understanding Quadratic Functions

Quadratic functions are a type of polynomial function represented by the equation $ f(x) = ax^2 + bx + c $. The distinct parabola-shaped curve they form on a graph makes them easy to identify. In the simplest form, $ f(x) = x^2 $, the graph is a symmetrical U-shaped curve opening upwards, centered at the origin (0, 0).
This curve represents the path where the y-values increase as you move away from the center along the x-axis. Changing the value of 'a' affects the width and direction of the parabola. A larger positive 'a' makes it narrower, while a negative 'a' flips it upside down.

Vertex - The highest or lowest point of a parabola. For $ f(x) = x^2 $, it is at the origin.
Axis of Symmetry - A vertical line passing through the vertex, dividing the graph into two identical halves.
Roots or Zeros - Points where the graph intersects the x-axis. In standard form, the roots of $ f(x) = ax^2 + bx + c $ can be found using the quadratic formula or factoring.

Exploring Function Transformations

Function transformations involve changing the basic form of a function, like a quadratic, to shift or skew it on a graph.
It's a crucial concept in understanding how mathematical functions can dynamically represent real-world scenarios or solve problems.
There are several types of transformations that can be applied to functions to achieve different effects:

**Horizontal Compress/Stretches** - Achieved by replacing $ x $ with a multiple, such as $ kx $. If $ |k| > 1 $, it results in horizontal compression. If $ 0 < |k| < 1 $, it causes horizontal stretching.
**Vertical Stretches/Shrinks** - Multiply the whole function by a constant factor. Larger coefficients stretch the graph, while smaller compress it.
**Reflections** - Changing the sign of $ y $ or $ x $ to mirror the graph across either the x-axis or y-axis.
**Translations/Slides** - Moving the graph without changing its shape. For example, replacing $ x $ with $ x + d $ shifts it horizontally, while adding a constant $ c $ to the function shifts it vertically.

Graph Shifting Simplified

Graph shifting is a straightforward transformation applied to functions to move their position on the coordinate plane. It doesn't change the shape or orientation of the graph; it merely relocates it to a new set of coordinates.
Using the problem context as an example, let's break down how graph shifting works through simple steps:

**Horizontal Shifting** - When you replace $ x $ with $ x - 5 $, it moves the graph to the right by 5 units. This is because each x-value is augmented by 5 before the function is evaluated.
**Vertical Shifting** - By adding 1 to the entire function, $ f(x) = 4(x-5)^2 + 1 $, it shifts the graph upwards by 1 unit. This is equivalent to moving the y-value of each point by 1.

Combining these shifts results in the graph of $ f(x) = 4(x - 5)^2 + 1 $, where the original shape is preserved but repositioned accordingly on the plane. These shifts are powerful tools in graphing, making it easy to manipulate the location of a function without altering its general appearance.

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91影视

Write a formula for \(f(x)=x^{2}\) horizontally compressed by a factor of \(\frac{1}{2},\) then shifted to the right 5 units and up 1 unit.

Short Answer

Step by step solution

Start with the Basic Function

Apply Horizontal Compression

Shift Right by 5 Units

Shift Up by 1 Unit

Key Concepts

Understanding Quadratic Functions

Exploring Function Transformations

Graph Shifting Simplified

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