/*! 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 3 Plot the following graphs on the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 3

Plot the following graphs on the same axes between the values \(x=-3\) to \(x=+3\) and determine the gradient and \(y\)-axis intercept of each. (a) \(y=3 x\) (b) \(y=3 x+7\) (c) \(y=-4 x+4\) (d) \(y=-4 x-5\)

Short Answer

Expert verified
(a) Gradient: 3, Y-intercept: 0; (b) Gradient: 3, Y-intercept: 7; (c) Gradient: -4, Y-intercept: 4; (d) Gradient: -4, Y-intercept: -5.

Step by step solution

01

Understanding Linear Equations

Each equation given is in the form of a straight line, represented by the general equation \(y = mx + c\), where \(m\) is the gradient and \(c\) is the \(y\)-intercept.
02

Finding the Gradient and Y-Intercept

For each equation, identify \(m\) (the coefficient of \(x\)) as the gradient and \(c\) (the constant term) as the \(y\)-intercept. (a) For \(y = 3x\): Gradient \(m = 3\), Y-intercept \(c = 0\). (b) For \(y = 3x + 7\): Gradient \(m = 3\), Y-intercept \(c = 7\). (c) For \(y = -4x + 4\): Gradient \(m = -4\), Y-intercept \(c = 4\). (d) For \(y = -4x - 5\): Gradient \(m = -4\), Y-intercept \(c = -5\).
03

Plotting the Graphs

To plot the graphs, select several values of \(x\) between \(-3\) and \(+3\). Calculate \(y\) for each equation using these \(x\) values and then plot the points on a Cartesian plane. Draw a straight line through the plotted points for each equation.
04

Analyzing Overlapping Graph Features

After plotting, observe each graph's gradient: Lines with positive gradient (\(y = 3x\) and \(y = 3x + 7\)) will rise as \(x\) increases, while those with negative gradient (\(y = -4x + 4\) and \(y = -4x - 5\)) will fall. Compare their \(y\)-intercepts to determine where each line crosses the \(y\)-axis.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Gradient Calculation
The gradient, also known as the slope of a line, indicates how steep a line is on a graph. It is determined by the change in the vertical direction (along the y-axis) divided by the change in the horizontal direction (along the x-axis). In a linear equation of the form \( y = mx + c\), the gradient is represented by the coefficient \(m\).
Here's how you can identify and interpret the gradient in each scenario:
  • Positive Gradient: A positive \(m\) value means that as you move along the x-axis, the line rises or goes upwards. For example, in the equations \(y = 3x\) and \(y = 3x + 7\), the gradient \(m = 3\) indicates a rising line.
  • Negative Gradient: A negative \(m\) value means that the line falls or goes downwards. In \(y = -4x + 4\) and \(y = -4x - 5\), the gradient \(m = -4\) shows a descending line.
Understanding the gradient helps in predicting how a line behaves as it moves along the graph.
Y-Intercept Identification
The y-intercept is the point where the line crosses the y-axis. It occurs when the value of x is zero. In the linear equation \(y = mx + c\), the y-intercept is represented by the constant term \(c\). It tells where the line will intersect the vertical axis of the graph.
For the given equations:
  • In \(y = 3x\), the y-intercept is \(c = 0\), meaning the line crosses the y-axis at the point (0, 0).
  • In \(y = 3x + 7\), the y-intercept is \(c = 7\), so the line crosses the y-axis at (0, 7).
  • In \(y = -4x + 4\), the y-intercept is \(c = 4\), crossing the y-axis at (0, 4).
  • In \(y = -4x - 5\), the y-intercept is \(c = -5\), where the line intersects at (0, -5).
Recognizing the y-intercept allows you to start drawing the graph from a known point, making plotting easier.
Graph Plotting
Plotting a graph involves drawing the line that corresponds to each equation between defined values, such as \(x = -3\) to \(x = +3\). Here鈥檚 a step-by-step guide to effectively plot these lines:
  • Choose Points: Select several x-values within the given range. Calculate the corresponding y-values using each equation.
  • Plot Points: Write down your calculated (x, y) pairs and plot them on the Cartesian plane.
  • Draw the Line: Connect the plotted points with a straight line. Ensure the line is extended and crosses the entire graph region within the given x-range.
When you correctly plot the graphs, you鈥檒l be able to see clearly how each equation behaves over the same set of axes.
Cartesian Plane
The Cartesian plane is a two-dimensional graphing system comprising a horizontal axis (x-axis) and a vertical axis (y-axis). It is named after Ren茅 Descartes, who developed this coordinate system. Key features of the Cartesian plane include:
  • Origin: The point (0, 0) where the x-axis and y-axis intersect.
  • Quadrants: The x and y-axes divide the plane into four quadrants, each representing a unique combination of positive and negative x and y values.
  • Horizontal and Vertical Lines: Horizontal lines run parallel to the x-axis, while vertical ones run parallel to the y-axis.
Understanding how to navigate the Cartesian plane is crucial for graphing equations like those provided in your exercise. This plane allows for precise placement of points based on their x and y coordinates, enabling a visual representation of mathematical equations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Plot the graph \(3 x+y+1=0\) and \(2 y-5=x\) on the same axes and find their point of intersection.

Without plotting graphs, determine the gradient and \(y\)-axis intercept values of the following equations: (a) \(y=7 x-3\) (b) \(3 y=-6 x+2\) (c) \(y-2=4 x+9\) (d) \(\frac{y}{3}=\frac{x}{3}-\frac{1}{5}\) (e) \(2 x+9 y+1=0\)

The temperature in degrees Celsius and the corresponding values in degrees Fahrenheit are shown in the table below. Construct rectangular axes, choose a suitable scale and plot a graph of degrees Celsius (on the horizontal axis) against degrees Fahrenheit (on the vertical scale). $$ \begin{array}{|l|llrrrr|} \hline{ }^{\circ} \mathrm{C} & 10 & 20 & 40 & 60 & 80 & 100 \\ { }^{\circ} \mathrm{F} & 50 & 68 & 104 & 140 & 176 & 212 \\ \hline \end{array} $$ From the graph find (a) the temperature in degrees Fahrenheit at \(55^{\circ} \mathrm{C}\), (b) the temperature in degrees Celsius at \(167^{\circ} \mathrm{F}\), (c) the Fahrenheit temperature at \(0^{\circ} \mathrm{C}\), and (d) the Celsius temperature at \(230^{\circ} \mathrm{F}\).

Determine the gradient of the straight line graph passing through the co- ordinates (a) \((-2,5)\) and \((3,4)\), and (b) \((-2,-3)\) and \((-1,3)\)

Experimental tests to determine the breaking stress \(\sigma\) of rolled copper at various temperatures \(t\) gave the following results $$ \begin{aligned} &\begin{array}{|l|ccc|} \hline \text { Stress } \sigma \mathrm{N} / \mathrm{cm}^{2} & 8.46 & 8.04 & 7.78 \\ \text { Temperature } t^{\circ} \mathrm{C} & 70 & 200 & 280 \\ \hline \end{array}\\\ &\begin{array}{|l|ccc|} \hline \text { Stress } \sigma \mathrm{N} / \mathrm{cm}^{2} & 7.37 & 7.08 & 6.63 \\ \text { Temperature } t^{\circ} \mathrm{C} & 410 & 500 & 640 \\ \hline \end{array} \end{aligned} $$ Show that the values obey the law \(\sigma=a t+b\), where \(a\) and \(b\) are constants and determine approximate values for \(a\) and \(b\). Use the law to determine the stress at \(250^{\circ} \mathrm{C}\) and the temperature when the stress is \(7.54 \mathrm{~N} / \mathrm{cm}^{2}\).
See all solutions

Recommended explanations on Physics Textbooks

Energy Physics

Read Explanation

Fundamentals of Physics

Read Explanation

Famous Physicists

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Mechanics and Materials

Read Explanation

Rotational Dynamics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.