Problem 13 - Single Variable Calculus

Question

In Exercises \(7-24\), sketch the graph of the function and find its absolute maximum and absolute minimum values, if any. $$ f(x)=x^{2}-4 x+3 \text { on }(-\infty, \infty) $$

Step-by-Step Solution

Answer

The function \(f(x) = x^2 - 4x + 3\) has a vertex at \((2, -1)\) and opens upward, which means it has an absolute minimum value but no absolute maximum value. The zeros of the function are at \(x = 3\) and \(x = 1\). The absolute minimum value is \(f(2) = -1\) at \(x = 2\).

Step 1: Identify the vertex of the parabola

To find the vertex, we'll use the formula \(h = \frac{-b}{2a}\) for the x-coordinate of the vertex, where \(f(x) = ax^2 + bx + c\). In our case, \(a = 1\), \(b = -4\), and \(c = 3\). So, we have \(h = \frac{-(-4)}{2(1)} = \frac{4}{2} = 2\). Now, we'll find the y-coordinate of the vertex by substituting the value of \(h\) into the function: \(k = f(2) = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1\) Thus, the vertex of the parabola is at the point \((2, -1)\).

Step 2: Determine direction of the parabola

Since the coefficient of the \(x^2\) term (or 'a') is positive, the parabola opens upward. This means that it has an absolute minimum value but no absolute maximum value.

Step 3: Find the zeros of the function

To find the zeros of the function, we set \(f(x) = 0\) and solve for \(x\): \(x^2 - 4x + 3 = 0\) We can factor this equation: \((x - 3)(x - 1) = 0\) Thus, the zeros of the function are \(x = 3\) and \(x = 1\).

Step 4: Locate absolute minimum value

Since we determined that the parabola opens upward and has a vertex at \((2, -1)\), the absolute minimum value of the function is \(f(2) = -1\) at \(x = 2\).

Step 5: Sketch the graph

Now that we have all the necessary information, we can sketch the graph of the function. 1. Plot the vertex: \((2, -1)\). 2. Plot the zeros: \((3, 0)\) and \((1, 0)\). 3. Draw the parabola opening upward, passing through the vertex and zeros. By following these steps, we have sketched the graph of the function and found its absolute minimum value. The function has an absolute minimum value of \(-1\) at \(x = 2\), but no absolute maximum value.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial feature as it represents the highest or lowest point on its curve, depending on the orientation. For quadratic functions such as \( f(x) = ax^2 + bx + c \), the vertex can be found using a simple formula for the x-coordinate: \( h = \frac{-b}{2a} \). This formula calculates the axis of symmetry, a vertical line that splits the parabola into two mirror-image halves.

In the given function \( f(x)=x^2-4x+3 \), we identify \( a = 1 \), \( b = -4 \), and \( c = 3 \). Plugging these into the formula gives \( h = \frac{4}{2} = 2 \).

Substitute \( x = 2 \) back into the function to compute the corresponding y-coordinate: \( k = f(2) = (2)^2 - 4(2) + 3 = -1 \). Therefore, the vertex is at the point \( (2, -1) \). This point reflects the curve’s lowest position for a parabola that opens upwards, providing us with critical information about its graph's shape and position.

Absolute Minimum and Maximum

Understanding absolute minimum and maximum values in a function helps us identify the extremities on a graph, peaks or valleys that the function reaches. For parabolas, this is relatively straightforward due to their symmetrical and predictable nature.

Since the parabola in our example opens upward, it has a vertex representing the absolute minimum. Notice here that the vertex at \( (2, -1) \) implies the lowest value the function can achieve is \(-1\) when \( x = 2 \).

On the contrary, an upward opening parabola lacks an absolute maximum because the arms of the parabola extend infinitely upward, meaning there isn't a highest point it settles at forever as \( x \) moves along the number line. Therefore, for this function, there is no absolute maximum, highlighting how the graph can rise beyond bounds but still possesses a definitive, calculated low point.

Zeros of a Function

Zeros, or roots, of a function are the x-values where the function equals zero. They are the points at which the graph intersects the x-axis. These values are essential for sketching the graph since they provide crucial data about where the curve crosses from positive to negative and vice versa.

To find the zeros for our function \( f(x) = x^2 - 4x + 3 \), we solve the equation \( x^2 - 4x + 3 = 0 \). By factoring, we rewrite this expression as \( (x - 3)(x - 1) = 0 \). Thus, the solutions are \( x = 3 \) and \( x = 1 \).

These zeros translate to the points \( (3, 0) \) and \( (1, 0) \) on the graph, denoting interception with the x-axis. Knowing these zeros is imperative in graphing the function accurately, ensuring the curve passes through key points while smoothly transitioning through its minimum or maximum.