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Chapter 3: Problem 13

Graph the given functions. $$y=x^{2}$$

Short Answer

Expert verified
Graph the parabolic curve using key points and ensure it is symmetrical around the y-axis with the vertex at the origin.

Step by step solution

01

Understand the Function

The given function is \( y = x^2 \), a quadratic function. Quadratic functions are represented as parabolas when graphed on a Cartesian plane. In this case, the parabola opens upwards since the coefficient of \( x^2 \) is positive.
02

Identify Key Features

For the function \( y = x^2 \), the vertex (the lowest point) is at the origin, (0,0). The axis of symmetry is the vertical line \( x = 0 \). The parabola has a symmetry about the \( y \)-axis.
03

Plot Points

Select several values of \( x \) to find corresponding \( y \) values. For example: When \( x = -2, y = 4 \); \( x = -1, y = 1 \); \( x = 0, y = 0 \); \( x = 1, y = 1 \); and \( x = 2, y = 4 \). Plot these points on the graph.
04

Draw the Parabola

Connect the plotted points smoothly to form the parabola. Ensure that it passes through each plotted point and is symmetrically curved around the \( y \)-axis.
05

Label the Graph

Label the x-axis and the y-axis, and provide an appropriate title for the graph. Clearly label the vertex and ensure the curve is accurately reflecting the parabolic shape described by the quadratic equation.

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Key Concepts

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

Understanding Quadratic Equations
Quadratic equations are a type of polynomial equation where the highest power of the variable is squared. They take the general form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants.
  • In these equations, \( a \) must not be zero; otherwise, it becomes a linear equation.
  • The graph of a quadratic function is known as a parabola.
  • Each quadratic equation has a vertex, which can be a maximum or minimum point.
When a quadratic equation is simplified into the form \( y = x^2 \) like in our exercise, it implies that:
  • The coefficient \( a \) is 1, so the graph opens upwards.
  • There is no linear or constant term; hence, the graph is symmetrical around the y-axis.
These fundamental properties help you predict the shape and position of the graph on a Cartesian plane.
Graphing on the Cartesian Plane
The Cartesian plane is a two-dimensional number line where all points are expressed with an x-coordinate and a y-coordinate.
  • It consists of a horizontal axis, called the x-axis, and a vertical axis, called the y-axis.
  • When graphing quadratic functions, points are plotted in the form (x, y), where both x and y are numerical values.
  • The plane is divided into four quadrants, allowing for both positive and negative values of x and y.
For the function \( y = x^2 \):
  • When you plot the points on this plane, the graph's symmetry becomes evident.
  • The vertex (0, 0) lies at the origin, making it a central reference point for the parabola.
  • Using specific values for \( x \) allows for constructing an accurate graph by plotting points such as (-2, 4), (-1, 1), (0, 0), (1, 1), and (2, 4).
Drawing these points precisely is crucial to achieving an accurate parabolic shape.
Exploring Parabolas
A parabola is the U-shaped graph you see when plotting a quadratic function, like \( y = x^2 \). This specific graph is a symmetrical curve.
  • For the equation \( y = ax^2 + bx + c \), if \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.
  • The vertex for \( y = x^2 \) is at the origin (0,0), which is also the point of symmetry.
  • Parabolas are always symmetrical along a vertical line called the axis of symmetry. For \( y = x^2 \), this line runs along \( x = 0 \).
The curvature of a parabola and the direction in which it opens contribute significantly to how it is interpreted in various contexts. In the graph of \( y = x^2 \), the curve smoothly passes through its plotted points, creating a consistent and predictable shape.

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Most popular questions from this chapter

Graph the indicated functions. The voltage \(V\) across a capacitor in a certain electric circuit for a \(2-s\) interval is \(V=2 t\) during the first second and \(V=4-2 t\) during the second second. Here, \(t\) is the time (in s). Plot \(V\) as a function of \(t\).

Graph the indicated functions. A guideline of the maximum affordable monthly mortgage \(M\) on a home is \(M=0.25(I-E),\) where \(I\) is the homeowner's monthly income and \(E\) is the homeowner's monthly expenses. If E= 600 dollar, graph \(M\) as a function of \(I\) for I= 2000 dollar to I= 10,000 dollar.

Solve the indicated equations graphically. Assume all data are accurate to two significant digits unless greater accuracy is given. In finding the illumination at a point \(x\) feet from one of two light sources that are \(100 \mathrm{ft}\) apart, it is necessary to solve the equation \(9 x^{3}-2400 x^{2}+240,000 x-8,000,000=0 .\) Find \(x\)

Graph the indicated functions. The consumption of fuel \(c\) (in \(\mathrm{L} / \mathrm{h}\) ) of a certain engine is determined as a function of the number \(r\) of \(\mathrm{r} / \mathrm{min}\) of the engine, to be \(c=0.011 r+40 .\) This formula is valid for \(500 \mathrm{r} / \mathrm{min}\) to \(3000 \mathrm{r} / \mathrm{min}\). Plot \(c\) as a function of \(r .(r\) is the symbol for revolution.)

Solve the given problems. A balloon is being blown up at a constant rate. (a) Sketch a reasonable graph of the radius of the balloon as a function of time. (b) Compare to a typical situation that can be described by \(r=\sqrt[3]{3 t},\) where \(r\) is the radius (in \(\mathrm{cm}\) ) and \(t\) is the time (in \(\mathrm{s}\) ).
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