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Chapter 2: Problem 32

Graph the equations by plotting points. $$ y=x^{2} $$

Short Answer

Expert verified
Plot points using the equation and then connect the points to draw the quadratic graph.

Step by step solution

01

- Understand the Equation

The given equation is a quadratic function: \[ y = x^2 \]This means that for each value of x, the corresponding y value is the square of x.
02

- Create a Table of Values

Choose a range of x values and calculate the corresponding y values. For example: \[\begin{array}{c|c}x & y \ \hline-3 & 9 \ -2 & 4 \ -1 & 1 \ 0 & 0 \ 1 & 1 \ 2 & 4 \ 3 & 9 \ \end{array}\]
03

- Plot the Points

On a graph, plot the points (-3, 9), (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), and (3, 9).
04

- Draw the Graph

Connect the points with a smooth, continuous curve to show the shape of the quadratic function y = x虏.

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Key Concepts

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

Plotting points
Plotting points on a graph is a simple yet vital step in visualizing mathematical functions. It's how we translate equations into shapes.
To plot a point, you need two coordinates: an x-value (horizontal position) and a y-value (vertical position).
For the equation \( y = x^2 \), you start by calculating the value of y for different values of x.
Each pair of x and y values forms a point like (x, y). For example, if x = 2, then y = 2^2 = 4, so you plot the point (2, 4).
Keep adding points to get a clearer picture of the function shape.
Quadratic equations
A quadratic equation is any equation in the form of \( y = ax^2 + bx + c \). It is characterized by the highest degree of 2 for the variable x.
The given equation, \( y = x^2 \), is a simple quadratic function with a = 1, b = 0, and c = 0.
Quadratic equations create U-shaped curves called parabolas. If a > 0, the parabola opens upwards. If a < 0, it opens downwards.
Recognizing a quadratic equation by its standard form helps you understand that its graph is symmetric and forms a parabola.
Creating a table of values
Creating a table of values is a systematic way to determine coordinates for plotting.
Start by choosing a range of x values that includes both negative and positive numbers. This ensures you cover all sections of the parabola.
For each chosen x-value, calculate the corresponding y-value using the equation \( y = x^2 \).
Let's say you choose x-values from -3 to 3. Calculate to find: \( (-3, 9), (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), (3, 9) \).
Organize these values in a table for easy reference when plotting on the graph.
Graphing step by step
Graphing a function step by step involves plotting points and then drawing the curve.
Start by labeling your graph with an x-axis (horizontal) and a y-axis (vertical). Mark equal intervals for both axes.
Next, use the table of values to plot each point. For example, plot the point (-3, 9) by moving 3 units left on the x-axis and 9 units up on the y-axis.
Continue plotting all points from your table.
Finally, connect the points with a smooth, continuous curve to form the parabola.
This visual representation makes it easier to understand the shape and behavior of the quadratic function.

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Most popular questions from this chapter

Refer to the functions \(f\) and \(g\) and evaluate the functions for the given values of \(x\). \(f=\\{(2,4),(6,-1),(4,-2),(0,3),(-1,6)\\} \quad\) and \(\quad g=\\{(4,3),(0,6),(5,7),(6,0)\\}\) $$(f \circ f)(-1)$$

Given \(m(x)=\sqrt{x-4},\) evaluate \((m \circ m)(x)\) and write the domain in interval notation.

Given \(h(x)=\frac{1}{x-6},\) evaluate \((h \circ h)(x)\) and write the domain in interval notation.

The amount of \(\mathrm{CO}_{2}\) emitted per year \(A(x)\) (in tons) for a vehicle that burns \(x\) miles per gallon of gas, can be approximated by \(A(x)=0.0092 x^{2}-0.805 x+21.9 .\) (Source: U.S. Department of Energy, http://energy.gov) a. Determine the difference quotient. \(\frac{A(x+h)-A(x)}{h}\) b. Evaluate the difference quotient on the interval \([20,25],\) and interpret its meaning in the context of this problem. c. Evaluate the difference quotient on the interval \([35,40],\) and interpret its meaning in the context of this problem.

Graph the function. $$ r(x)=\left\\{\begin{array}{cl} x^{2}-4 & \text { for } x \leq 2 \\ 2 x-4 & \text { for } x>2 \end{array}\right. $$
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