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Chapter 2: Problem 62

Use transformations to graph the functions. $$ u(x)=-(x-1)^{2}-2 $$

Short Answer

Expert verified
Shift right 1 unit, reflect across the x-axis, then move down 2 units.

Step by step solution

01

- Identify the Base Function

Identify the base function. In this case, the base function is the quadratic function: \(f(x) = x^2\).
02

- Apply Horizontal Shift

Observe the horizontal shift in the function \(u(x) = -(x-1)^2 - 2\). The expression \(x-1\) inside the parentheses indicates a horizontal shift to the right by 1 unit.
03

- Apply Vertical Shift

Consider the vertical shift. The term \(-2\) outside the parentheses indicates a vertical shift downward by 2 units.
04

- Apply Reflection

The negative sign in front of the quadratic term \(-(x-1)^2\) indicates a reflection across the x-axis.
05

- Combine All Transformations

Combine all identified transformations. Start with the base graph \(y = x^2\): shift it one unit to the right, reflect it across the x-axis, and then shift it 2 units downward. This results in the graph of \(u(x) = -(x-1)^2 - 2\).

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Key Concepts

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

quadratic function
A quadratic function is a type of polynomial where the highest degree of any term is 2. The general form of a quadratic function is given by: \[ f(x) = ax^2 + bx + c \]In simpler terms, the basic quadratic function is often written as \( f(x) = x^2 \). This function produces a U-shaped curve known as a parabola. When graphed, a quadratic function opens upward if the coefficient of \( x^2 \) is positive, and it opens downward if the coefficient is negative.
horizontal shift
A horizontal shift involves moving a graph left or right along the x-axis. For the function \[ u(x) = -(x-1)^2 - 2 \]the expression \( x-1 \) inside the parentheses tells us there is a horizontal shift. Specifically, it indicates a shift to the right by 1 unit. This is because the standard form \( (x-h) \) results in a shift to the right when \( h \) is positive and to the left when \( h \) is negative.
vertical shift
A vertical shift moves a graph up or down along the y-axis. In our function \[ u(x) = -(x-1)^2 - 2 \]the term \( -2 \) outside the parentheses indicates a vertical shift downward. The entire graph of the function is lowered by 2 units. This is because a positive constant moved the graph up and a negative constant moves it down.
reflection across x-axis
A reflection across the x-axis flips the graph upside down. For transformations in a quadratic function like \[ u(x) = -(x-1)^2 - 2 \]the negative sign before the squared term \( -(x-1)^2 \) indicates such a reflection. This means any point \( (x, y) \) on the graph of the base function \( f(x) = x^2 \) will be flipped to \( (x, -y) \), effectively inverting the parabola along the x-axis.

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Most popular questions from this chapter

A function is given. (See Examples \(4-5)\) a. Find \(f(x+h)\). b. Find \(\frac{f(x+h)-f(x)}{h}\). $$f(x)=x^{2}-3 x$$

Use a graphing utility to graph the piecewise-defined function. $$ \begin{aligned} &z(x)=\left\\{\begin{array}{ll} 2.5 x+8 & \text { for } x<-2 \\ -2 x^{2}+x+4 & \text { for }-2 \leq x<2 \\ -2 & \text { for } x \geq 2 \end{array}\right.\\\ &\text { Is there actually a "gap" in the graph at } x=2 ? \end{aligned} $$

Use a graphing utility to graph the piecewise-defined function. $$ \begin{aligned} &k(x)=\left\\{\begin{array}{ll} -2.7 x-4.1 & \text { for } x \leq-1 \\ -x^{3}+2 x+5 & \text { for }-1

Determine if the function is even, odd, or neither. $$ k(x)=13 x^{3}+12 x $$

Refer to the functions \(r, p,\) and \(q .\) Evaluate the function and write the domain in interval notation. \(r(x)=-3 x \quad p(x)=x^{2}+3 x \quad q(x)=\sqrt{1-x}\) $$(r \cdot q)(x)$$
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