slope
Algebra
Calculus
(noun)
also called gradient; slope or gradient of a line describes its steepness
Examples of slope in the following topics:
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Slope
- Slope is often denoted by the letter $m$.
- In other words, a line with a slope of $-9$ is steeper than a line with a slope of $7$.
- The slope of the line is $\frac{4}{5}$.
- We can see the slope is decreasing, so be sure to look for a negative slope.
- Calculate the slope of a line using "rise over run" and identify the role of slope in a linear equation
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Slope-Intercept Equations
- Writing an equation in slope-intercept form is valuable since from the form it is easy to identify the slope and $y$-intercept.
- Let's write the equation $3x+2y=-4$ in slope-intercept form and identify the slope and $y$-intercept.
- Now that the equation is in slope-intercept form, we see that the slope $m=-\frac{3}{2}$, and the $y$-intercept $b=-2$.
- The value of the slope dictates where to place the next point.
- The slope is $2$, and the $y$-intercept is $-1$.
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Point-Slope Equations
- The point-slope equation is another way to represent a line; only the slope and a single point are needed.
- Given a slope, $m$, and a point $(x_{1}, y_{1})$, the point-slope equation is:
- Example: Write the equation of a line in point-slope form, given a point $(2,1)$ and slope $-4$, and convert to slope-intercept form
- Since we have two points, but no slope, we must first find the slope:
- Plug this point and the calculated slope into the point-slope equation to get:
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Parallel and Perpendicular Lines
- Recall that the slope-intercept form of an equation is: $y=mx+b$ and the point-slope form of an equation is: $y-y_{1}=m(x-x_{1})$, both contain information about the slope, namely the constant $m$.
- This means that if the slope of one line is $m$, then the slope of its perpendicular line is $\frac{-1}{m}$.
- The two slopes multiplied together must equal $-1$.
- Also, the product of the slopes equals $-1$.
- Again, start with the slope-intercept form and substitute the values, except the value for the slope will be the negative reciprocal.
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Slope and Y-Intercept of a Linear Equation
- For the linear equation y = a + bx, b = slope and a = y-intercept.
- What is the y-intercept and what is the slope?
- The slope is 15 (b = 15).
- (a) If b > 0, the line slopes upward to the right.
- (c) If b < 0, the line slopes downward to the right.
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Slope and Intercept
- The concepts of slope and intercept are essential to understand in the context of graphing data.
- The slope or gradient of a line describes its steepness, incline, or grade.
- A higher slope value indicates a steeper incline.
- Slope is normally described by the ratio of the "rise" divided by the "run" between two points on a line.
- It also acts as a reference point for slopes and some graphs.
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The Derivative and Tangent Line Problem
- The use of differentiation makes it possible to solve the tangent line problem by finding the slope $f'(a)$.
- The slope of the secant line passing through $p$ and $q$ is equal to the difference quotient
- If $k$ is known, the equation of the tangent line can be found in the point-slope form:
- It barely touches the curve and shows the rate of change slope at the point.
- Define a derivative as the slope of the tangent line to a point on a curve
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Derivatives and Rates of Change
- Thus, to solve the tangent line problem, we need to find the slope of a line that is "touching" a given curve at a given point, or, in modern language, that has the same slope.
- But what exactly do we mean by "slope" for a curve?
- In this case, $y = f(x) = m x + b$, for real numbers m and b, and the slope m is given by:
- This gives an exact value for the slope of a straight line.
- If $x$ and $y$ are real numbers, and if the graph of $y$ is plotted against $x$, the derivative measures the slope of this graph at each point.
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Direction Fields and Euler's Method
- Direction fields, also known as slope fields, are graphical representations of the solution to a first order differential equation.
- The slope field is traditionally defined for differential equations of the following form:
- An isocline (a series of lines with the same slope) is often used to supplement the slope field.
- Then, from the differential equation, the slope to the curve at $A_0$ can be computed, and thus, the tangent line.
- Along this small step, the slope does not change too much $A_1$ will be close to the curve.
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Demand Curve
- The demand curve in a monopolistic competitive market slopes downward, which has several important implications for firms in this market.
- The demand curve of a monopolistic competitive market slopes downward.
- The demand curve for an individual firm is downward sloping in monopolistic competition, in contrast to perfect competition where the firm's individual demand curve is perfectly elastic.
- Because the individual firm's demand curve is downward sloping, reflecting market power, the price these firms will charge will exceed their marginal costs.
- As you can see from this chart, the demand curve (marked in red) slopes downward, signifying elastic demand.