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Chapter 9: Problem 297

Graph each function using a vertical shift. $$g(x)=x^{2}+2$$

Short Answer

Expert verified
The graph of \( g(x) = x^2 + 2 \) is a parabola shifted 2 units up from the graph of \( f(x) = x^2 \).

Step by step solution

01

- Understand the Basic Graph

First, recognize that the base function before applying any shifts is the quadratic function: \( f(x) = x^2 \). The graph of \( f(x) = x^2 \) is a parabola that opens upwards with its vertex at the origin (0,0).
02

- Identify the Vertical Shift

Next, observe the term that indicates the vertical shift. In this exercise, the function is \( g(x) = x^2 + 2 \), which adds a constant value of 2 to the original function \( f(x) = x^2 \). This indicates a vertical shift of +2 units.
03

- Shift the Graph Vertically

Apply the vertical shift by moving every point on the original graph of \( f(x) = x^2 \) up by 2 units. Specifically, this means if a point on the graph of \( f(x) = x^2 \) has coordinates \( (x, y) \), the corresponding point on the graph of \( g(x) = x^2 + 2 \) will have coordinates \( (x, y+2) \).
04

- Plot the New Graph

Plot the new points obtained from the vertical shift. The new graph will be a parabola that looks similar to \( f(x) = x^2 \) but shifted 2 units up. The vertex of the new graph \( g(x) = x^2 + 2 \) is at (0, 2).

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Key Concepts

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

Quadratic Functions
Quadratic functions are one of the fundamental concepts in algebra. They have the general form:
\( f(x) = ax^2 + bx + c \).
In this equation:
  • \( a \) determines the direction of the parabola (upward or downward).
  • \( b \) affects the symmetry and position of the vertex along the x-axis.
  • \( c \) indicates the y-intercept, where the graph crosses the y-axis.
Most commonly, we deal with parabola-shaped graphs opening either upwards or downwards.
A positive \( a \) value (like in \( f(x) = x^2 \)) represents upwards opening, while a negative value flips it downwards.
Vertical Shifts in Graphs
Vertical shifts are transformations that move a graph up or down. They do not affect the shape of the graph, only its position.
If we add a constant \( k \) to a function, it results in a vertical shift. For example, the function \( g(x) = f(x) + k \), will move the original function \( f(x) \) up by \( k\) units if \( k \) is positive, or down by \( k \) units if negative.
In our specific example, \( g(x)=x^2+2 \), the graph of \( x^2 \) is shifted upwards by 2 units.
This is because the added constant (2) moves every point on the graph up by that amount.
Parabolas
Parabolas are U-shaped curves associated with quadratic functions. The main features of a parabola include:
  • **Vertex**: The highest or lowest point depending on the direction.
  • **Axis of Symmetry**: A vertical line passing through the vertex, dividing the parabola into two mirror images.
  • **Direction**: Determined by the sign of \( a \). Upward if \( a > 0\) and downward if \(-a < 0\).
In the quadratic function \(f(x) = x^2 \), the vertex is at the origin (0,0).
For the function \( g(x) = x^2 + 2 \), the parabola shifts upward, moving the vertex to (0, 2).
This vertical shift doesn鈥檛 change the curvature or direction, only the graph鈥檚 position.

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Most popular questions from this chapter

(a) graph the quadratic functions on the same rectangular coordinate system and (b) describe what effect adding a constant, \(k_{t}\) to the function has on the basic parabola. \(f(x)=x^{2}, g(x)=x^{2}+7\) and \(h(x)=x^{2}-7\).

(a) graph the quadratic functions on the same rectangular coordinate system and (b) describe what effect adding a constant, \(k_{t}\) to the function has on the basic parabola. \(f(x)=x^{2}, g(x)=x^{2}+4\) and \(h(x)=x^{2}-4\).

Solve. Round answers to the nearest tenth. A computer store owner estimates that by charging \(x\) dollars each for a certain computer, he can sell 40 - \(x\) computers each week. The quadratic function \(R(x)=-x^{2}+40 x\) is used to find the revenue, \(R,\) received when the selling price of a computer is \(x\) Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue.

(a) solve graphically and (b) write the solution in interval notation. $$x^{2}+6 x+5>0$$

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it using properties. $$f(x)=-2 x^{2}-4 x-5$$
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