Which Steps Should Be Used To Graph The Equation Below?$ Y - 4 = \frac{1}{3}(x + 2) $A. 1. Plot The Point $ (2, 4) $.2. From That Point, Count Left 3 Units And Down 1 Unit, And Plot A Second Point.3. Draw A Line Through The Two
Understanding the Equation
The given equation is in the form of a linear equation, which is a polynomial equation of degree one. The general form of a linear equation is:
y = mx + b
where m is the slope and b is the y-intercept.
The given equation is:
y - 4 = \frac{1}{3}(x + 2)
To graph this equation, we need to follow a series of steps.
Step 1: Find the y-Intercept
The y-intercept is the point where the line intersects the y-axis. To find the y-intercept, we need to set x = 0 and solve for y.
y - 4 = \frac{1}{3}(0 + 2) y - 4 = \frac{2}{3} y = 4 + \frac{2}{3} y = \frac{14}{3}
So, the y-intercept is at the point (0, \frac{14}{3}).
Step 2: Find the Slope
The slope is the rate of change of the line. To find the slope, we need to look at the coefficient of x in the equation.
y - 4 = \frac{1}{3}(x + 2)
The coefficient of x is \frac{1}{3}, which is the slope.
Step 3: Plot the Point
To plot the point, we need to find a point on the line. We can use the y-intercept as the point.
Plot the point (0, \frac{14}{3}).
Step 4: Count Left 3 Units and Down 1 Unit
To count left 3 units and down 1 unit, we need to move 3 units to the left and 1 unit down from the point (0, \frac{14}{3}).
The new point is (0 - 3, \frac{14}{3} - 1) = (-3, \frac{11}{3}).
Step 5: Plot the Second Point
Plot the point (-3, \frac{11}{3}).
Step 6: Draw a Line Through the Two Points
Draw a line through the two points (0, \frac{14}{3}) and (-3, \frac{11}{3}).
Alternative Method: Using the Equation
We can also graph the equation by using the equation itself.
y - 4 = \frac{1}{3}(x + 2)
To graph this equation, we can use the following steps:
- Plot the point (2, 4).
- From that point, count left 3 units and down 1 unit, and plot a second point.
- Draw a line through the two points.
This method is a bit more intuitive and easier to understand.
Conclusion
Frequently Asked Questions
Q: What is the difference between a linear equation and a non-linear equation? A: A linear equation is a polynomial equation of degree one, while a non-linear equation is a polynomial equation of degree two or higher.
Q: How do I find the y-intercept of a linear equation? A: To find the y-intercept, set x = 0 and solve for y.
Q: What is the slope of a linear equation? A: The slope is the rate of change of the line, which is the coefficient of x in the equation.
Q: How do I plot a point on a linear equation? A: To plot a point, use the y-intercept as the point and then count left or right and up or down to find the second point.
Q: What is the difference between a horizontal line and a vertical line? A: A horizontal line is a line that is parallel to the x-axis, while a vertical line is a line that is parallel to the y-axis.
Q: How do I graph a linear equation using the equation itself? A: To graph a linear equation using the equation itself, plot the point (2, 4), count left 3 units and down 1 unit, and plot a second point, and then draw a line through the two points.
Q: What is the significance of the slope-intercept form of a linear equation? A: The slope-intercept form of a linear equation (y = mx + b) is significant because it allows us to easily identify the slope and y-intercept of the line.
Q: How do I determine if a linear equation is a function or not? A: A linear equation is a function if it passes the vertical line test, meaning that no vertical line intersects the graph of the equation at more than one point.
Q: What is the difference between a linear equation and a quadratic equation? A: A linear equation is a polynomial equation of degree one, while a quadratic equation is a polynomial equation of degree two.
Q: How do I graph a linear equation using a graphing calculator? A: To graph a linear equation using a graphing calculator, enter the equation into the calculator and use the graphing function to visualize the line.
Q: What are some common mistakes to avoid when graphing linear equations? A: Some common mistakes to avoid when graphing linear equations include:
- Not using the correct slope and y-intercept
- Not plotting the correct points
- Not drawing the line through the correct points
- Not using the correct graphing function on a graphing calculator
Conclusion
Graphing linear equations can be a bit challenging, but by following the steps outlined above and avoiding common mistakes, we can easily graph the equation. We can also use the equation itself to graph the line and use a graphing calculator to visualize the line.