Tangent lines​​ with zero slope


a. Graph the function f(x)=x^2−4x+3.

Verified step by step guidance

1. <Step 1: Understand the problem.> We need to graph the function $f(x)=x^2-4x+3$ and identify where the tangent lines have a zero slope.
2. <Step 2: Find the derivative.> The derivative of $f(x)$, denoted as $f^{\prime }(x)$, gives us the slope of the tangent line at any point $x$. Calculate $f^{\prime }(x)$ using the power rule.
3. <Step 3: Set the derivative to zero.> To find where the tangent line has a zero slope, set $f^{\prime }(x)=0$ and solve for $x$.
4. <Step 4: Solve for x.> Solve the equation from Step 3 to find the x-coordinate(s) where the slope of the tangent line is zero.
5. <Step 5: Graph the function.> Plot the function $f(x)=x^2-4x+3$ and mark the point(s) found in Step 4 on the graph. These are the points where the tangent line is horizontal.>

Key Concepts

Tangent Lines

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point. Understanding tangent lines is crucial for analyzing the behavior of functions, especially in relation to their derivatives.

Derivative

The derivative of a function at a point quantifies how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function as the interval approaches zero. For the function f(x) = x^2 - 4x + 3, finding the derivative will help identify points where the slope of the tangent line is zero, indicating potential local maxima or minima.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient 'a'. Analyzing the graph of f(x) = x^2 - 4x + 3 will reveal its vertex, intercepts, and the nature of its tangent lines.