Question

Graph the parabola \(f(x)=x^{2} .\) Explain why the secant lines between the points \((-a, f(-a))\) and \((a, f(a))\) have zero slope. What is the slope of the tangent line at \(x=0 ?\)

Short answer

Question: Prove that all secant lines of the parabola \(f(x) = x^2\) passing through points \((-a, f(-a))\) and \((a, f(a))\) have a zero slope and determine the slope of the tangent line at \(x=0\). Answer: The slope of all secant lines passing through points \((-a, f(-a))\) and \((a, f(a))\) of the parabola \(f(x) = x^2\) is zero, which makes these lines horizontal. The slope of the tangent line at \(x=0\) is also zero, implying that the tangent line is horizontal at the vertex of the parabola.

Step-by-step solution

Graph the parabola

To graph the parabola \(f(x) = x^2\), we can create a table of values for \(x\) and \(f(x)\). Use the equation \(f(x) = x^2\) to find the corresponding y-values. Next, plot these points and connect them to form the graph of the parabola. It's important to notice that the graph of \(f(x)=x^2\) is symmetric with respect to the y-axis.

Analyze the secant line between points \((-a, f(-a))\) and \((a, f(a))\)

Given two points \((-a,f(-a))\) and \((a,f(a))\), we can find the slope of the secant line using the formula: $$ m = \frac{f(a) - f(-a)}{a - (-a)} $$ Recall that the function is \(f(x) = x^2\). So, we have: $$ f(-a) = (-a)^2 = a^2 $$ and $$ f(a) = a^2 $$

Prove the secant lines have zero slope

Using the values of \(f(-a)\) and \(f(a)\) from the previous step, we can now find the slope of the secant line: $$ m = \frac{f(a) - f(-a)}{a - (-a)} = \frac{a^2 - a^2}{a - (-a)} = \frac{0}{2a} = 0 $$ The slope of the secant lines is 0, which means that these lines are horizontal. This is because the parabola \(f(x)=x^2\) is symmetric with respect to the y-axis and the given points are equidistant from the axis of symmetry.

Find the slope of the tangent line at \(x=0\)

To find the slope of the tangent line at \(x=0\), we'll need to compute the derivative of the function \(f(x)\): $$ f'(x) = \frac{d}{dx}(x^2) = 2x $$ Now, evaluate the derivative at \(x = 0\) to find the slope of the tangent line at that point: $$ f'(0) = 2 (0) = 0 $$ The slope of the tangent line at \(x=0\) is also 0, meaning that the tangent line is horizontal at the vertex of the parabola.

Key concepts

Secant Lines Slope

Understanding the slope of a secant line is crucial in grasping the behavior of a function between two points. In the context of parabolas, particularly the function f(x) = x^2, the secant line serves as an average rate of change over an interval.

When we examine the secant lines connecting the points (-a, f(-a)) and (a, f(a)) on the graph of our parabola, we are looking at a line that stretches over the axis of symmetry. Due to the symmetric nature of the parabola, both points have the same 'height' since f(-a) = (-a)^2 = a^2 = f(a). Thus, when calculating the slope m of this secant line, the change in y (the numerator) is zero, resulting in a zero slope.

This horizontal line that appears due to the symmetry on either side of the y-axis tells us that the average rate of change between these symmetric points is zero—there is no net rise or fall on this secant.

Tangent Line Slope

The slope of the tangent line to a curve at a given point provides an instantaneous rate of change at that exact point. For the function f(x) = x^2, the slope of the tangent line at the point x=0 holds special significance. It gives us information about how the function is changing at the vertex of the parabola.

The tangent line at x=0 is found by taking the derivative of f(x), which gives us f'(x) = 2x. At x=0, the derivative evaluates to zero, meaning that the instantaneous rate of change of the parabola at the vertex is zero. This leads to the tangent line being horizontal at that point. In simpler terms, at the very tip of the parabola’s vertex, the curve is neither increasing nor decreasing—it's at a momentary standstill.

Derivative of a Function

The derivative of a function represents the concept of instantaneous rate of change, akin to finding the slope of a tangent line at any point on a curve. It's a fundamental tool in calculus for analyzing how functions behave.

For any given function like f(x) = x^2, the process of finding its derivative encapsulates calculating the limit of the slope of secant lines as the interval they span approaches zero. In practice, this means finding a formula that gives the slope of the tangent line at any point x on the function.

Specifically, the derivative of f(x) = x^2 is f'(x) = 2x. This comes from the power rule, which states that the derivative of x^n is nx^(n-1). Therefore, at any point x on our parabola, the slope of the tangent line is double the x-coordinate, which aligns with the steepness or slope of the curve at that specific point.