Slope of a Line Consider the equation π = ππ β π We can make an π, π chart to help graph the line represented by this equation. π π π βπ π π π π π π Do you notice a pattern between the change of π values and the change of π values? This pattern is referred to as the slope of the line. The slope of a line is the ratio of the change in π (vertical change) to the change in π (horizontal change). We use the letter π to represent slope. π= ππππππ ππ π ππππππ ππ π 1 Letβs look at: π = ππ β π +π The π values increase by 1 +π +π π= π π π βπ π π π π π π +π +π The π values increase by 2 +π ππππππ ππ π π = =π ππππππ ππ π π therefore, π = π for the line π = ππ β π. We can use the slope to help graph the line represented by the equation π = ππ β π. +1 +2 +1 +2 2 Looking left to right: If the line is going up, the line has a negative slope If the line is going down, the line has a positive slope Slope is zero Undefined slope All horizontal lines have a slope equal to zero. All vertical lines have an undefined slope. 3 Finding the slope of a line given two points on the line: Letβs call π·1 (point π) (ππ , ππ ) and π·2 (point π) (π2 , π2 ) π ππ β ππ π2 π·π ππ β ππ ππ π·π π π2 ππ π = ππππππ ππ π βπ ππ β ππ = = ππππππ ππ π βπ ππ β ππ Example 1 Find the slope of the line that passes through the points (π, π) and (ππ, π). ππ β ππ πβπ π= = ππ β ππ ππ β π = = *reduce 4 Parallel Lines: Two lines are parallel if they never intersect. Parallel lines have the same slope. Perpendicular Lines: Two lines are perpendicular if and only if they intersect at a ππ° angle. The slopes of two perpendicular lines are negative reciprocals of each other. ππ π2 π1 π π ππ π β π π π π πΌππ ππππππ π β Finding the slope using the equation of a line: Recall: Equation of a line π = ππ + π Step 1: Solve the equation for π Step 2: identify the coefficient of π β this is the slope of the line. 5 Example 2 Find the slope of the line ππ + π = π Step 1: ππ + π = π βππ β ππ π = βππ + π NOTE: we prefer to see the π before the constant Step 2: The coefficient of π is β π Therefore, the slope of the line 2π₯ + π¦ = 8 is β π. NOTE: Any line parallel to the line ππ + π = π has a slope of β π π ands any line perpendicular to the line has a slope of . π 6 Example 3 Identify the lines as being parallel, perpendicular or neither: π+π=π ππ + ππ = π First we find the slope of each line. Step 1: Solve for π (for both Line 1 and Line 2) Line 1: Line 2: π+π=π ππ + ππ = π βπ βπ βππ β ππ π = βπ + π ππ = βππ + π Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ π π π Slope of line 1: ππ = βπ π = βπ + π Slope of line 2: ππ = βπ Therefore, these two lines are parallel 7 Slope of a Line Practice Problems 1. Find the slope of the line that passes through the points (π, π) and (β π, π). 2. Match each equation with its corresponding graph: π: a) π + π = π π¦ π₯ ππ: b) β π + π = π π¦ π₯ πππ: π¦ c) π = π π₯ ππ: π¦ π₯ d) π = π 8
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