Explain in detail where the formula for the difference quotient comes from now that you know! Include all appropriate terminology (secant line, tangent line, h/delta x, etc). Your post must include text and some form of media (picture/video) to support.
There is a opened-up parabola line that has a secant line that touches the graph twice. So the first point that intersect with the graph will have the x-value equal to x and y-value equals to f(x) (because there is no specific number for the value, it is better if the values replace by the variables). To prove that it is correct, I will make a tangent line that touches the graph exactly at that first point. The point can only be defined due to the value based on the parabola graph. So that point should be the same as the first point of the secant line. For the second point, I will name h for the difference between the first and second points on the x-axis. Because the second point has x-value equal to x+h, the y-value changes as well, it turns into f(x+h). Next, I will find the slope for the secant line by using m= (y2 - y1) / (x2 - x1). Although it is weird to use the variable instead of number, the result is still accurate. So I will plug in m= (f(x+h) - f(x))/((x+h)-x). The x variables on the denominator will cancel out. Notice that f(x+h) - f(x) cannot be simplified because the two function are not the same. So the difference quotient is the slope of the secant line. The secant line is related to the tangent line. Because tangent line only touches the graph once, there is no h in the slope. Due to the nonexistence of the value of h, h will be come 0. It explains why I have to find the limit as h approaches 0. As h approaches 0, the secant line approaches the tangent line.
From math.bu.edu
An example of difference quotient (include a image)
From IntuitiveMath
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