/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 Sketch each equation. $$ p(t... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 18

Sketch each equation. $$ p(t)=-2+3 t $$

Short Answer

Expert verified
The graph of \( p(t) = -2 + 3t \) is a straight line with a y-intercept at \(-2\) and slope of 3, passing through points (0,-2) and (1,1).

Step by step solution

01

Identify Type of Equation

The equation given is in linear form: \( p(t) = -2 + 3t \). This is a classic slope-intercept form equation \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept.
02

Determine the Y-Intercept

Set \( t = 0 \) to find the y-intercept. When \( t = 0 \), the equation becomes \( p(0) = -2 + 3(0) = -2 \). Thus, the y-intercept is \( -2 \). Plot the point \((0, -2)\) on the graph.
03

Use Slope to Find Another Point

The slope \( m \) is 3, which means for every 1 unit increase in \( t \), \( p(t) \) increases by 3 units. From point \((0, -2)\), move 1 unit to the right and 3 units up to find another point \((1, 1)\). Plot this point.
04

Draw the Line

With the points \((0, -2)\) and \((1, 1)\) plotted, draw a straight line through these points. This line represents the equation \( p(t) = -2 + 3t \). Use a ruler to ensure that the line is straight.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a fundamental concept in understanding linear equations. It is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) represents the y-intercept. The slope-intercept form makes it easy to graph equations and understand how different values change the shape and position of the line.
  • The slope \( m \) indicates the steepness or incline of the line.
  • The y-intercept \( b \) shows where the line crosses the y-axis.
This form is extremely useful because it provides a clear starting point and direction for sketching a line on a coordinate plane.
Graphing Linear Equations
Graphing linear equations involves representing the equation visually on a coordinate plane. In slope-intercept form, \( y = mx + b \), graphing becomes straightforward:
  • Identify the y-intercept \( b \).
  • Use the slope \( m \) to determine another point on the line.
Once these points are plotted, connect them with a straight line. Always extend the line through your points to show the entire scope of the equation. You can use graph paper or a digital graphing tool to ensure accuracy.The key steps for graphing include:
  • Plot the y-intercept on the y-axis.
  • Use the slope to calculate the next point.
  • Draw the line through these points.
This straightforward process helps visualize how changes to \( m \) or \( b \) will shift or rotate the line.
Y-Intercept
The y-intercept \( b \) is the point where the line crosses the y-axis. It is crucial because it provides the starting point for graphing the line. To find the y-intercept from the equation \( y = mx + b \), simply set \( x = 0 \). For example, in the equation \( p(t) = -2 + 3t \), setting \( t = 0 \) gives us \( p(0) = -2 \). This means the line crosses the y-axis at \((0, -2)\). Understanding the y-intercept:
  • It is always the constant term \( b \) in the expression \( y = mx + b \).
  • Provides a fixed starting point for the line on the graph.
Plotting the y-intercept carefully ensures accuracy in graphing.
Slope
The slope \( m \) of a linear function determines the angle and direction of the line. It is found in the equation \( y = mx + b \) as the coefficient of the variable. The slope represents how much \( y \) changes for each unit increase in \( x \). For the function \( p(t) = -2 + 3t \), the slope is \( 3 \), indicating:
  • For every increase of 1 in \( t \), \( p(t) \) increases by 3.
  • A positive slope means the line rises as it moves to the right.
  • A negative slope would imply the line falls to the right.
To visualize this, use the slope to move from the y-intercept along the graph. It helps in understanding the direction and pace at which the line changes. This gives clarity in predicting the behavior of the equation, without needing to plot numerous points.

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