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Chapter 2: Problem 34

\(f(x)=(x+6)^2+3\); horizontal shrink by a factor of \(\frac{1}{2}\) and a translation 1 unit down, followed by a reflection in the \(x\)-axis

Short Answer

Expert verified
After performing all the given transformations on \(f(x)=(x+6)^2+3\), we get the new function as \(f(x) = -(2x + 12)^2 - 2\).

Step by step solution

01

Horizontal Shrink

Apply the horizontal shrink to the function by multiplying the \(x\) term inside the brackets, before the squaring, by 2 which is the reciprocal of 1/2. This gives us the new function \(f(x) = (2(x + 6))^2 + 3\) or \(f(x) = (2x + 12)^2 + 3\).
02

Vertical Translation

Translate the function vertically by subtracting 1 from the entire function. This gives us \(f(x) = (2x + 12)^2 + 3 - 1\) or \(f(x) = (2x + 12)^2 + 2\).
03

Reflection in the x-axis

Reflect the function in the x-axis by multiplying the entire function by -1. This gives us \(f(x) = -((2x + 12)^2 + 2)\) or \(f(x) = -(2x + 12)^2 - 2\).

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Key Concepts

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

Horizontal Shrink
In transformations of functions, a horizontal shrink compresses the graph towards the y-axis. This occurs when the x-values are multiplied by a factor greater than 1.
A horizontal shrink by a factor of \(\frac{1}{2}\) involves multiplying the \(x\) term by 2 before any transformations, since \(\frac{1}{2}\) is the reciprocal of 2. In mathematical notation, this modifies the expression \((x + 6)\) in the given function, turning it into \((2(x + 6))\) or \((2x + 12)\).
This transformation squeezes the graph into half of its original width without altering its shape. Thus, the new function after applying the horizontal shrink becomes \(f(x) = (2x + 12)^2 + 3\).
Vertical Translation
Vertical translation involves shifting the entire graph of a function up or down along the y-axis. It occurs by adding or subtracting a constant from the entire function.
In this specific exercise, the function is moved 1 unit down. This is achieved by subtracting 1 from the entire function \(f(x)\). The function transforms from \((2x + 12)^2 + 3\) to \((2x + 12)^2 + 2\).
  • Subtracting a positive number moves the graph down.
  • Adding a positive number shifts the graph up.
This operation changes the location of the graph vertically but doesn鈥檛 alter its shape or structure.
Reflection in the x-axis
A reflection in the x-axis is a type of transformation that flips the graph of a function upside down.
Mathematically, this is achieved by multiplying the entire function by -1. This inverts the output values (the y-values) of the function.
For the previous vertical translation step, the function was \(f(x) = (2x + 12)^2 + 2\). After reflecting in the x-axis, the function becomes \(f(x) = -((2x + 12)^2 + 2)\) or \(f(x) = -(2x + 12)^2 - 2\).
  • Reflection transforms points \((x, y)\) to \((x, -y)\).
  • This affects the entire graph symmetrically about the x-axis.
Such a transformation results in the graph looking like a mirror image of the original graph, flipped over the x-axis.
Quadratic Functions
Quadratic functions form a vital part of algebra and are represented in the standard form as \(f(x) = ax^2 + bx + c\).
These functions create a parabolic curve called a parabola, which can open upwards or downwards depending on the leading coefficient \(a\).
In the given exercise, the expression \((x + 6)^2 + 3\) originally represents a quadratic function. During transformations鈥攈orizontal shrink, vertical shift, reflection鈥攖he function remains quadratic but appears differently graphically.
  • If \(a > 0\), the parabola opens upwards.
  • If \(a < 0\), the parabola opens downwards.
These transformations don鈥檛 change the fact that the function is quadratic; they just change its orientation and position on the coordinate plane.

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Most popular questions from this chapter

Describe a real-life situation that can be modeled by a quadratic equation. Justify your answer.

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PROBLEM SOLVING A woodland jumping mouse hops along a parabolic path given by y = ?0.2x2 + 1.3x, where x is the mouse鈥檚 horizontal distance traveled (in feet) and y is the corresponding height (in feet). Can the mouse jump over a fence that is 3 feet high? Justify your answer.

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