Question

What is the slope of the secant line between the points \((a, f(a))\) and \((b, f(b))\) on the graph of \(f ?\)

Short answer

Answer: The slope of the secant line between the points (a, f(a)) and (b, f(b)) on the graph of function f is given by the formula: Slope = (f(b) - f(a)) / (b - a).

Step-by-step solution

Identify the coordinates of the given points

For point 1: \(x_1 = a\) and \(y_1 = f(a)\). For point 2: \(x_2 = b\) and \(y_2 = f(b)\).

Calculate the change in x (\(\Delta x\))

To calculate the change in x between the two points, subtract the x-coordinates: \(\Delta x = x_2 - x_1 = b - a\).

Calculate the change in y (\(\Delta y\))

To calculate the change in y between the two points, subtract the y-coordinates: \(\Delta y = y_2 - y_1 = f(b) - f(a)\).

Calculate the slope of the secant line

The slope of the secant line is the ratio between the change in y and the change in x. Use the following formula: \(\text{slope} = \frac{\Delta y}{\Delta x}\). Plug in the values calculated in steps 2 and 3: \(\text{slope} = \frac{f(b) - f(a)}{b - a}\).

Final Answer:

The slope of the secant line between the points \((a, f(a))\) and \((b, f(b))\) on the graph of \(f\) is \(\frac{f(b) - f(a)}{b - a}\).

Key concepts

Secant Line

A secant line is a straight line that connects two points on the graph of a function. This is not the same as a tangent line, which touches the graph at just one point without crossing it. Instead, the secant line passes through the curve between two distinct points.
The primary purpose of a secant line is to estimate the average rate of change of the function between these two points. This can be visualized as the slope of the secant line. In essence, the secant line gives us a broader look at how the function behaves over an interval. The more closely placed the two points are, the more accurately the slope of the secant line can approximate the slope of the tangent line at one of those points.

Change in x (Δx)

The change in x, denoted as ( Δx ), represents the difference in the x-coordinates of the two points involved. It is essentially how much movement there is horizontally between these points on the graph.
To find ( Δx ), you subtract the x-coordinate of the first point from the x-coordinate of the second point, given by the formula: ( Δx = x_2 - x_1 = b - a ). This measurement is crucial when calculating the slope of the secant line, as it forms the denominator in the slope formula. Without knowing ( Δx ), we can't correctly assess how steeply the change happens across the x-axis.

Change in y (Δy)

Change in y, or ( Δy ), indicates the difference in the y-coordinates of the two points on the function's graph. It measures how much vertical change occurs between these points, showing how the function's output varies over this interval.
The formula to determine ( Δy ) is straightforward: subtract the first y-coordinate from the second, expressed as ( Δy = y_2 - y_1 = f(b) - f(a) ). This value captures the total rise or fall from one point to the other. Like ( Δx ), ( Δy ) is essential for determining the slope of the secant line because it constitutes the numerator in the slope formula.

Graph of a Function

A graph of a function is a visual representation of how the function's outputs relate to its inputs. Typically plotted on a Cartesian plane, the horizontal axis represents the input values (x), and the vertical axis represents the output values (y).
By plotting points defined by these x and y values, the graph forms a curve or line that exhibits the behavior of the function. Each point on the graph corresponds to an input-output pair ((x, y) = (x, f(x)) ).
Understanding a function's graph allows us to visually interpret various properties, such as continuity, minimum and maximum points, and the overall trend of growth or decay. The secant line drawn on this graph between any two points provides insight into the average changes between those points. It offers a means to quantify the rate at which the function is increasing or decreasing over that interval.