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Chapter 0: Problem 4

Graph each pair of equations on one set of axes. $$y=x^{2} \text { and } y=x^{2}-3$$

Short Answer

Expert verified
Graph \( y = x^2 \) and then graph \( y = x^2 - 3 \) by shifting it down 3 units.

Step by step solution

01

- Understand the given equations

The given equations are: 1. \( y = x^2 \) 2. \( y = x^2 - 3 \). Both are quadratic equations that graph as parabolas.
02

- Identify vertex and orientation

For the equation \( y = x^2 \):- The vertex is at (0, 0).- The parabola opens upwards.For the equation \( y = x^2 - 3 \):- The vertex is at (0, -3).- The parabola also opens upwards.
03

- Plotting key points for \( y = x^2 \)

Choose some values for x and compute their corresponding y-values for \( y = x^2 \): - When \( x = -2, y = 4 \) (giving the point (-2, 4))- When \( x = -1, y = 1 \)- When \( x = 0, y = 0 \)- When \( x = 1, y = 1 \)- When \( x = 2, y = 4 \)Plot these points and sketch the parabola.
04

- Plotting key points for \( y = x^2 - 3 \)

Choose some values for x and compute their corresponding y-values for \( y = x^2 - 3 \):- When \( x = -2, y = 1 \) (giving the point (-2, 1))- When \( x = -1, y = -2 \)- When \( x = 0, y = -3 \)- When \( x = 1, y = -2 \)- When \( x = 2, y = 1 \)Plot these points and sketch the parabola.
05

- Compare the graphs

Observe that both parabolas open upwards. The graph of \( y = x^2 - 3 \) is a vertical shift of the graph of \( y = x^2 \) downwards by 3 units.

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Key Concepts

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

y = x^2
The equation \( y = x^2 \) represents a basic quadratic function. This equation forms a curve called a parabola. In this specific case, the parabola is oriented upwards.

Here are some key points to understand about \( y = x^2 \):
  • The vertex, or the turning point, of the parabola is at (0, 0).
  • When x is positive or negative, squaring it will always produce a non-negative y value.
  • The graph is symmetric about the y-axis because \( (-x)^2 = x^2 \).
To sketch this graph, you can plot several key points where x takes some values such as -2, -1, 0, 1, and 2:
  • When \( x = -2 \), \( y = 4 \)
  • When \( x = -1 \), \( y = 1 \)
  • When \( x = 0 \), \( y = 0 \)
  • When \( x = 1 \), \( y = 1 \)
  • When \( x = 2 \), \( y = 4 \)
Plot these points on a graph and connect them with a smooth curve to form the parabola.
vertical shift
A vertical shift refers to moving the entire graph of a function up or down on the coordinate plane. This shift does not affect the shape of the graph, only its position.

For the equation \( y = x^2 - 3 \), the graph is a vertical shift of the basic quadratic equation \( y = x^2 \). Here's how it works:
  • The \( -3 \) means that every y-value of the original parabola \( y = x^2 \) is decreased by 3.
  • This moves the vertex from (0, 0) down to (0, -3).
To plot the shifted graph, use the same x-values as before and compute the new y-values:
  • When \( x = -2 \), \( y = 4 - 3 = 1 \)
  • When \( x = -1 \), \( y = 1 - 3 = -2 \)
  • When \( x = 0 \), \( y = 0 - 3 = -3 \)
  • When \( x = 1 \), \( y = 1 - 3 = -2 \)
  • When \( x = 2 \), \( y = 4 - 3 = 1 \)
This will give you a parabola that looks identical in shape to \( y = x^2 \) but is shifted 3 units downwards.
parabolas
Parabolas are U-shaped curves that graph quadratic equations. They are defined by an equation in the form \( y = ax^2 + bx + c \). Here are some essential features of parabolas:
  • Vertex: The highest or lowest point of the parabola (depending on whether it opens up or down).
  • Axis of Symmetry: A vertical line that passes through the vertex and divides the parabola into two mirror-image halves.
  • Direction: Parabolas can open upwards (if the coefficient of \( x^2 \) is positive) or downwards (if it is negative).
In the equations \( y = x^2 \) and \( y = x^2 - 3 \), both parabolas open upwards. Here's why grasping these concepts is important:
  • It helps to predict the shape and direction of the graph.
  • Understanding the vertex allows you to quickly identify the maximum or minimum point.
  • Knowing the effects of shifts (horizontal and vertical) helps in transforming and sketching graphs.
The comparison between \( y = x^2 \) and \( y = x^2 - 3 \) showcases a vertical shift. While the parabolas maintain their shape, the second graph is moved downwards, showing the importance of shifts in graphing functions.

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