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Chapter 1: Problem 13

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=\sqrt{x}\) appropriately, and then use a graphing utility to confirm that your sketch is correct. $$y=3-\sqrt{x+1}$$

Short Answer

Expert verified
The graph of \(y = 3-\sqrt{x+1}\) starts at (-1,3), decreasing to the right.

Step by step solution

01

Parent Function

The parent function for this equation is \(y = \sqrt{x}\). This is the standard square root function, which starts at the origin (0,0) and extends indefinitely to the right, forming a curved line that increases at a decreasing rate.
02

Horizontal Translation

The term \(x + 1\) in\(y = 3-\sqrt{x+1}\) indicates a horizontal translation. Specifically, \(y= \sqrt{x+1}\) indicates a shift of the parent function 1 unit to the left. Therefore, the starting point of the square root function shifts from (0,0) to (-1,0).
03

Vertical Reflection

The negative sign in front of the square root, \(-\sqrt{x+1}\), implies a reflection in the x-axis. This means that the curve of the square root function is flipped upside down; it will now decrease as it moves to the right.
04

Vertical Translation

The number 3 added to the equation corresponds to a vertical translation. In the equation \(y = 3-\sqrt{x+1}\), the entire function is shifted 3 units upwards. This means every point on the reflected curve is moved up by 3 units.
05

Combined Transformations

Combine all the transformations: The graph of \(y = \sqrt{x}\) is moved left by 1 unit (because of \(+1\)), reflected over the x-axis (because of the negative sign), and then shifted up by 3 units. The result is the graph of \(y = 3-\sqrt{x+1}\), starting at (-1,3) and moving downward to the right.
06

Graphing Utility Confirmation

Use a graphing utility or graphing software to check the accuracy of your transformations. Input \(y = 3-\sqrt{x+1}\) into the graphing tool and verify that the graph starts at (-1, 3), decreases to the right, and resembles an upside-down square root shifted up by 3 units.

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Key Concepts

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

Horizontal Translation
Horizontal translation involves shifting a graph left or right from its original position. This is determined by modifications inside the function's parentheses. For instance, the expression \( x+1 \) inside the square root \( \sqrt{x+1} \) indicates a move to the left by 1 unit. Here鈥檚 how to identify this shift:
  • If you see \( x+c \), it shifts the graph left by \( c \) units.
  • If you see \( x-c \), it shifts the graph right by \( c \) units.
By translating \( y=\sqrt{x} \) to \( y=\sqrt{x+1} \), the graph moves its starting point from (0, 0) to (-1, 0). Translating horizontally doesn't affect the graph's height; it only changes its left-right position.
Vertical Reflection
Vertical reflection is a transformation where the graph is flipped over the x-axis. This happens when a negative sign is in front of a function. In our example, \( -\sqrt{x+1} \) reflects the square root function over the x-axis:
  • Originally, \( y=\sqrt{x} \) progresses upward as \( x \) increases.
  • With \( -\sqrt{x+1} \), the graph changes direction and now decreases as \( x \) increases.
Think of vertical reflection as turning the graph "upside down," which means any positive values become negative, making the function descend instead of climb.
Vertical Translation
Vertical translation shifts a function graph up or down. It is easy to identify because it's added or subtracted directly from the whole function. In \( y=3-\sqrt{x+1} \), the number 3 denotes moving the graph 3 units up:
  • Add \( c \): Moves the graph up by \( c \) units.
  • Subtract \( c \): Moves the graph down by \( c \) units.
After reflecting and translating our parent function, moving it 3 units up means each point on the graph is lifted higher, keeping the curve's shape but changing its vertical position.
Parent Function
A parent function is the simplest form of any set of functions that form a "family." For square-root functions, the parent is \( y=\sqrt{x} \). This foundational format is key to understanding how transformations change a graph:
  • It starts at (0, 0), growing gently upwards and to the right.
  • Every transformation (translation, reflection) is applied to this base form.
By recognizing the parent function, you can foresee how variations like \( y=3-\sqrt{x+1} \) will manifest, making it easier to understand and apply graph transformations successfully.

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Most popular questions from this chapter

Suppose that you begin cutting wedge-shaped pieces from a pie so that the are length along the outer crust of each piece is equal to the radius. What fraction of the pie will remain after all pieces that can be cut in this way are eaten?

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