Example Question - graphing a line
Here are examples of questions we've helped users solve.
Graphing a Line from Point-Slope Form Equation
To graph the line given by the equation \( y - 2 = \frac{-2}{5}(x - 4) \), you can follow these steps: 1. Recognize the equation form: The equation is in point-slope form, \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope of the line. 2. Identify the slope and a point: From the given equation, the slope (\( m \)) is \( -\frac{2}{5} \) and the point (\( x_1, y_1 \)) is (4, 2). 3. Plot the given point: Start by plotting the point (4, 2) on the coordinate plane. This is a point through which the line will pass. 4. Use the slope to find another point: The slope \( -\frac{2}{5} \) means that for every 5 units you move to the right (in the positive x-direction), you move 2 units down (in the negative y-direction) because the slope is negative. You could also move 5 units to the left (negative x-direction) and 2 units up (positive y-direction) to find another point. 5. Plot the second point: For example, starting at point (4, 2) and moving 5 units to the right, you reach the point (9, 0). Plot this point on the coordinate plane. 6. Draw the line: Once you have these two points plotted, you can use a ruler to draw a straight line through them, extending it across the coordinate plane. This line represents all the points that satisfy the equation. Here's a rough step-by-step description of plotting the points and drawing the line: - Place a dot at (4, 2). - From (4, 2), move 5 units to the right to get to (9, 2). - From (9, 2), move 2 units down to get to (9, 0). - Place a dot at (9, 0). - Connect the two dots with a straight line. - Extend the line through and beyond both points to indicate that it continues infinitely in both directions. If you're graphing this by hand or using a graphing tool, make sure to label your axes and scale appropriately to accurately depict the slope and points.
Graphing a Line Using Point-Slope Form
To graph the line given by the equation \( y - 2 = \frac{-2}{5}(x - 4) \), you can follow these steps: 1. **Identify the slope and y-intercept:** The equation is in point-slope form, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. In this case, the slope (\( m \)) is \( \frac{-2}{5} \) and the point on the line is (4, 2). 2. **Plot the point on the graph:** Begin by plotting the point (4, 2) on your graph paper. This will be a point through which the line passes. 3. **Use the slope to find another point:** The slope \( \frac{-2}{5} \) means you move 2 units down and 5 units to the right from the point (4, 2) to find another point on the line. You could also move 2 units up and 5 units to the left if you wish to go in the opposite direction. 4. **Draw the line:** Connect the two points with a straight line. This line will extend infinitely in both directions. It's important to be precise and to plot points correctly. If you have graph paper, use a ruler to ensure that the line passes through the points accurately and extends across the grid.
Graphing a Line in Slope-Intercept Form
To graph the line given by the equation: \[ y + 3 = \frac{9}{4}(x + 4) \] We should first rewrite it into the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Let's solve the equation for \( y \): \[ y = \frac{9}{4}(x + 4) - 3 \] Now, distribute the \( \frac{9}{4} \) across \( (x + 4) \): \[ y = \frac{9}{4}x + \frac{9}{4} \cdot 4 - 3 \] Simplify the constants: \[ y = \frac{9}{4}x + 9 - 3 \] \[ y = \frac{9}{4}x + 6 \] Now we have the equation in slope-intercept form with a slope \( m = \frac{9}{4} \) and a y-intercept \( b = 6 \). To graph this line, follow these steps: 1. Start by plotting the y-intercept (0,6) on the graph. 2. From this point, use the slope to determine the next point. The slope is \( \frac{9}{4} \) which means that for every 4 units you move to the right (in the positive x-direction), you move 9 units up (in the positive y-direction). 3. Plot another point using the slope. For example, starting at (0,6), go right 4 units to (4,6) and then up 9 units to (4,15). 4. Draw a straight line through the points to complete the graph of the line. Remember, you can plot more points if needed to ensure accuracy before drawing your line.
Graphing a Line in Slope-Intercept Form
Based on the image provided, the equation of the line is \(y + 3 = \frac{9}{4}(x + 4)\). To graph this line, it's typically easiest to write it in slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. Let's rewrite the given equation in slope-intercept form: \[ \begin{align*} y + 3 &= \frac{9}{4}(x + 4) \\ y &= \frac{9}{4}x + \frac{9}{4} \cdot 4 - 3 \\ y &= \frac{9}{4}x + 9 - 3 \\ y &= \frac{9}{4}x + 6 \end{align*} \] Now we have the equation in slope-intercept form, with a slope \(m=\frac{9}{4}\) and a y-intercept \(b=6\). To graph this line: 1. Start at the y-intercept (0,6) on the graph. 2. Use the slope to find another point. The slope is the rise over the run, meaning you go up 9 units and right 4 units from the y-intercept (or down 9 and left 4, for a point in the opposite direction). 3. Plot the second point. 4. Draw a line through the two points, extending it across the graph. This straight line represents the equation \(y = \frac{9}{4}x + 6\).
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