Problem 58
Question
Graph each equation in a rectangular coordinate system. \(3 x+12-0\)
Step-by-Step Solution
Verified Answer
The graph of the equation \(3x + 12\) is a straight line with a slope of \(3\) and y-intercept of \(12\).
1Step 1: Identify the Slope and Y-Intercept
The equation provided is \(3x + 12\). This is in the form \(y = mx + b\), where \(m = 3\) (the coefficient of \(x\)) is the slope and \(b = 12\) is the y-intercept.
2Step 2: Plot the Y-Intercept
Begin by drawing a rectangular coordinate system or a graph. Mark the y-intercept on the y-axis. This is the point where the line crosses the y-axis, which in this case is \(12\).
3Step 3: Use the Slope to Draw the Line
The slope (\(3\)) means that for each unit increase in \(x\), \(y\) increases by \(3\). Starting from the y-intercept, move to the right one unit and up three units and plot a point. Repeat this process until sufficient points are plotted. Draw the line that passes through all these points.
Key Concepts
Understanding Slope in Linear GraphsThe Role of Y-Intercept in GraphingExploring the Rectangular Coordinate System
Understanding Slope in Linear Graphs
The concept of slope is vital when graphing linear equations. The slope of a line in the equation form \(y = mx + b\) is represented by \(m\). It tells us how steep or flat the line is.
- If the slope is positive, the line slopes upwards as it moves from left to right.
- A negative slope means the line slopes downwards.
- A slope of zero indicates a horizontal line, meaning there's no change in \(y\) as \(x\) changes.
Let's consider a line with a slope of 3. This means that for every increase of 1 unit in \(x\) (moving right), \(y\) increases by 3 units (moving up). This makes the line quite steep. To visualize this, imagine starting at any point on the y-axis. Move over 1 space to the right (positive \(x\)-direction), then climb up 3 spaces. Plot a point there. Draw your line through these points to see the equation come to life.
Understanding how slope affects direction and steepness helps in plotting linear equations accurately.
- If the slope is positive, the line slopes upwards as it moves from left to right.
- A negative slope means the line slopes downwards.
- A slope of zero indicates a horizontal line, meaning there's no change in \(y\) as \(x\) changes.
Let's consider a line with a slope of 3. This means that for every increase of 1 unit in \(x\) (moving right), \(y\) increases by 3 units (moving up). This makes the line quite steep. To visualize this, imagine starting at any point on the y-axis. Move over 1 space to the right (positive \(x\)-direction), then climb up 3 spaces. Plot a point there. Draw your line through these points to see the equation come to life.
Understanding how slope affects direction and steepness helps in plotting linear equations accurately.
The Role of Y-Intercept in Graphing
The y-intercept is the point where your graph crosses the y-axis. In the equation \(y = mx + b\), \(b\) represents the y-intercept. This y-intercept is significant because it provides a starting point for graphing your line.
To identify the y-intercept, simply locate \(b\) in your equation. In our exercise, it is 12. On the graph, this is the point \((0, 12)\), meaning when \(x\) is zero, \(y\) is 12. Every line will have one y-intercept. It's like a launching pad from which the line emanates using the slope to determine its path.
When you begin graphing a line, you first plot the y-intercept. From this point, you use the slope to find additional points, helping to draw the full line. This base point ensures your graph is initialized correctly, ensuring accuracy in the resulting plotted equation.
To identify the y-intercept, simply locate \(b\) in your equation. In our exercise, it is 12. On the graph, this is the point \((0, 12)\), meaning when \(x\) is zero, \(y\) is 12. Every line will have one y-intercept. It's like a launching pad from which the line emanates using the slope to determine its path.
When you begin graphing a line, you first plot the y-intercept. From this point, you use the slope to find additional points, helping to draw the full line. This base point ensures your graph is initialized correctly, ensuring accuracy in the resulting plotted equation.
Exploring the Rectangular Coordinate System
A rectangular coordinate system, or Cartesian plane, consists of two perpendicular number lines intersecting at the origin \((0,0)\). The horizontal line is known as the x-axis while the vertical line is the y-axis. These axes split the plane into four quadrants, used to locate points uniquely with coordinates \((x, y)\).
To graph linear equations like \(y = 3x + 12\), we use this system to plot points.
Understanding the rectangular coordinate system is fundamental for graphing not just lines, but any mathematical function, making it an essential tool in mathematics.
To graph linear equations like \(y = 3x + 12\), we use this system to plot points.
- First, plot the y-intercept on the y-axis.
- Then use the slope to find other points by moving right or left along the x-axis and adjusting the y-coordinate accordingly.
Understanding the rectangular coordinate system is fundamental for graphing not just lines, but any mathematical function, making it an essential tool in mathematics.
Other exercises in this chapter
Problem 58
Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$ x^{2}+y^{2}+12 x-6 y-4=0
View solution Problem 58
Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$ for the given function. $$f(x)=6 x+1$$
View solution Problem 58
Find a. \((f \circ g)(x)\) b. \((g \circ f)(x)\) c. \((f \circ g)(2)\) d. \((g \circ f)(2)\) $$ f(x)-x^{2}+1, g(x)=x^{2}-3 $$
View solution Problem 59
Begin by graphing the standard quadratic function, \(f(x)-x^{2} .\) Then use transformations of this graph to graph the given function. $$ g(x)-(x-2)^{2} $$
View solution