Graphing in Slope-Intercept Form: A method of graphing linear equations using the y-intercept as a starting point and the slope to determine the direction and steepness of the line
- Identify the y-intercept \((0, b)\) from the equation \(y = mx + b\)
- Plot the y-intercept on the y-axis
- Identify the slope \(m\) and write it as a fraction \(\frac{\text{rise}}{\text{run}}\)
- From the y-intercept, use the slope to find another point: move up/down (rise) and right/left (run)
- Draw a straight line through the points
From \(y = 2x + 3\), the y-intercept is \(b = 3\), so the point is \((0, 3)\)
The slope is \(m = 2\), which can be written as \(\frac{2}{1}\) (rise = 2, run = 1)
Plot the point \((0, 3)\) on the y-axis
From \((0, 3)\), move up 2 units and right 1 unit to reach \((1, 5)\)
Draw a straight line through \((0, 3)\) and \((1, 5)\)
The line passes through \((0, 3)\) and \((1, 5)\) with slope \(m = 2\).
• Y-intercept Location: Always occurs at \((0, b)\)
• Slope as Fraction: Write as \(\frac{\text{rise}}{\text{run}}\) for graphing
• Positive Slope: Line rises from left to right
Negative Slope Graphing: When the slope is negative, the line falls from left to right, requiring downward movement when following the slope
From \(y = -\frac{1}{2}x + 4\), the y-intercept is \(b = 4\), so the point is \((0, 4)\)
The slope is \(m = -\frac{1}{2}\), which means \(\frac{-1}{2}\) (rise = -1, run = 2)
Plot the point \((0, 4)\) on the y-axis
From \((0, 4)\), move down 1 unit and right 2 units to reach \((2, 3)\)
Draw a straight line through \((0, 4)\) and \((2, 3)\)
The line passes through \((0, 4)\) and \((2, 3)\) with slope \(m = -\frac{1}{2}\).
• Negative Slope: Line falls from left to right
• Downward Movement: Negative rise means move down
• Graphing Direction: Always move right for positive run
Fractional Slope Graphing: When the slope is a fraction, the numerator represents the rise (vertical movement) and the denominator represents the run (horizontal movement)
From \(y = \frac{3}{4}x - 2\), the y-intercept is \(b = -2\), so the point is \((0, -2)\)
The slope is \(m = \frac{3}{4}\), which means rise = 3 and run = 4
Plot the point \((0, -2)\) on the y-axis
From \((0, -2)\), move up 3 units and right 4 units to reach \((4, 1)\)
Draw a straight line through \((0, -2)\) and \((4, 1)\)
The line passes through \((0, -2)\) and \((4, 1)\) with slope \(m = \frac{3}{4}\).
• Fractional Slope: Numerator = rise, denominator = run
• Graphing with Fractions: Use rise and run to move to next point
• Positive Fraction: Line rises from left to right
Slope-Intercept Form: A linear equation written as \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept
Y-intercept: The point where the line crosses the y-axis, occurring when \(x = 0\)
Slope: The rate of change, indicating steepness and direction of the line
- Identify Components: Extract \(m\) (slope) and \(b\) (y-intercept) from \(y = mx + b\)
- Plot Y-intercept: Mark the point \((0, b)\) on the y-axis
- Convert Slope: Write slope as fraction \(\frac{\text{rise}}{\text{run}}\)
- Find Next Point: From y-intercept, move according to rise and run
- Draw Line: Connect points with a straight line
Integer Slope: When the slope is a whole number like -1, it can be written as a fraction \(\frac{-1}{1}\) to show rise and run clearly
From \(y = -x + 1\), the y-intercept is \(b = 1\), so the point is \((0, 1)\)
The slope is \(m = -1\), which can be written as \(\frac{-1}{1}\) (rise = -1, run = 1)
Plot the point \((0, 1)\) on the y-axis
From \((0, 1)\), move down 1 unit and right 1 unit to reach \((1, 0)\)
Draw a straight line through \((0, 1)\) and \((1, 0)\)
The line passes through \((0, 1)\) and \((1, 0)\) with slope \(m = -1\).
• Integer Slope: Write as fraction over 1 for clarity
• Negative Integer: Means move down for positive run
• Graphing Consistency: Always move right for positive run
Linear Model Graphing: Real-world situations represented by linear equations, where the y-intercept represents the initial value and the slope represents the rate of change
From \(y = 1.5x + 10\), the y-intercept is \(b = 10\), so the point is \((0, 10)\)
The slope is \(m = 1.5\), which can be written as \(\frac{3}{2}\) (rise = 3, run = 2)
Plot the point \((0, 10)\) on the y-axis
From \((0, 10)\), move up 3 units and right 2 units to reach \((2, 13)\)
Draw a straight line through \((0, 10)\) and \((2, 13)\)
Y-intercept (10): Base cost is $10 even for 0 hours
Slope (1.5): Cost increases by $1.50 per hour
The line passes through \((0, 10)\) and \((2, 13)\) with slope \(m = 1.5\). The base cost is $10 and it increases by $1.50 per hour.
• Real-World Context: Y-intercept = initial value, Slope = rate of change
• Variable Identification: Determine dependent and independent variables
• Decimal to Fraction: Convert 1.5 to 3/2 for easier graphing
Slope-Intercept Form: A linear equation written as \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept.
Y-intercept (b): The value of \(y\) when \(x = 0\), the point where the line crosses the y-axis.
Slope (m): The rate of change, indicating how much \(y\) changes when \(x\) increases by 1.
Graphing Process: Using the y-intercept as a starting point and the slope to determine the direction of the line.
- Equation Analysis: Identify \(m\) (slope) and \(b\) (y-intercept) from \(y = mx + b\)
- Y-intercept Plotting: Plot the point \((0, b)\) on the y-axis
- Slope Conversion: Write slope as fraction \(\frac{\text{rise}}{\text{run}}\)
- Point Finding: From y-intercept, move according to rise and run to find additional points
- Line Drawing: Connect points with a straight line extending in both directions
- Verification: Check that the line has the correct steepness and direction
• Y-intercept Location: Always at point \((0, b)\)
• Slope Direction: Positive = rises, Negative = falls
• Graphing Movement: Rise over run determines next point location
• Linear Consistency: Any point on the line satisfies the equation
• Verification: Check that multiple points satisfy the equation