Problem 29 - Applied Calculus

Question

Graph each function "by hand." [Note: Even if you have a graphing calculator, it is important to be able to sketch simple curves by finding a few important points.] $$ f(x)=-3 x^{2}+6 x+9 $$

Step-by-Step Solution

Answer

The graph is a downward-opening parabola with vertex at \((1, 12)\).

Step 1: Identify the Function Type

The function given is a quadratic function of the form \( ax^2 + bx + c \), where \( a = -3 \), \( b = 6 \), and \( c = 9 \). Quadratic functions graph as parabolas.

Step 2: Determine the Vertex

The vertex of a parabola \( ax^2 + bx + c \) is given by the formula \( \left( \frac{-b}{2a}, f\left(\frac{-b}{2a}\right) \right) \). Calculate \( x = \frac{-6}{2(-3)} = 1 \). Substitute \( x = 1 \) into the function: \( f(1) = -3(1)^2 + 6(1) + 9 = 12 \). Thus, the vertex is at \( (1, 12) \).

Step 3: Determine the Axis of Symmetry

The axis of symmetry for the parabola is the vertical line through the vertex, which is \( x = 1 \).

Step 4: Identify Additional Points

Choose points on both sides of the vertex to plot. Let's calculate \( f(0) \), \( f(2) \), and another point if necessary. \( f(0) = 9 \) and \( f(2) = 9 \). These points give us a symmetric cross-sectional view across the axis of symmetry.

Step 5: Sketch the Parabola

Plot the vertex \( (1, 12) \) and points \( (0, 9) \) and \( (2, 9) \) on the coordinate grid. Connect these points in a smooth curve to form a U-shaped parabola, opening downwards since \( a = -3 \) is negative.

Key Concepts

Parabola

A parabola is a U-shaped curve that represents the graph of a quadratic function. You can always identify it when you see the equation in the form of \( ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants. When sketching a graph, the parabola will appear as a symmetrical curve. Its direction, whether opening upwards or downwards, is determined by the coefficient \( a \). If \( a \) is positive, the parabola opens upwards, like a happy face. If \( a \) is negative, like in our example \( f(x) = -3x^2 + 6x + 9 \), it opens downwards, like a sad face.
To effectively sketch a parabola, you need to determine its key features, like the vertex and axis of symmetry. These components will help you understand the shape and position of the parabola on a graph. It's important to start with these foundational steps before moving on to sketching.

Vertex

The vertex of a parabola is its highest or lowest point, depending on the direction it opens. For the function \( f(x) = ax^2 + bx + c \), the vertex is given by the coordinates \( \left( \frac{-b}{2a}, f\left(\frac{-b}{2a}\right) \right) \). Let's break this down:

• The first part of the vertex coordinates, \( \frac{-b}{2a} \), represents the x-coordinate. It tells us where the peak or trough of the parabola lies horizontally.
• The second part, \( f\left(\frac{-b}{2a}\right) \), gives the y-coordinate, or how high or low the vertex is.

In our example with \( a = -3 \), \( b = 6 \), the vertex calculates to \( (1, 12) \). This means the parabola reaches its maximum point at \( x = 1 \) with a height of 12. Being mindful of this point helps in plotting the curve accurately and understanding its maximum or minimum value.

Axis of Symmetry

The axis of symmetry is a crucial feature of parabolas. It is an imaginary vertical line that divides the parabola into two mirror-image halves. For the function \( ax^2 + bx + c \), it's governed by the x-coordinate of the vertex, \( x = \frac{-b}{2a} \).

• This line helps establish symmetry in the graph, as the parabola's two sides reflect evenly across it.
• In our function \( f(x) = -3x^2 + 6x + 9 \), the axis of symmetry is at \( x = 1 \).

This axis is fundamental when sketching because it guides you in determining other points on the parabola. Notice how any point \( (x, y) \) on one side of the axis will have a symmetric point at \((-x, y)\) on the other side. Understanding this helps in accurately plotting the graph and ensuring the curve remains balanced and accurately represented.

Sketching Graphs

Sketching graphs is the practical phase of understanding a quadratic function. It's the stage where calculations and theory come to life on paper. Here’s how you can effectively sketch a parabola:

• Start by plotting the vertex, which acts as a base for the graph.
• Next, mark the axis of symmetry to maintain balance in your sketch.
• Use additional calculated points on both sides of the vertex, such as \( f(0), f(2) \), or more if necessary, to give a clearer picture of how the graph unfolds.

In our example, with points like \( (0, 9) \) and \( (2, 9) \) added, the symmetry and shape of the parabola become obvious. Connect these points smoothly to form a U-shaped curve. Since \( a = -3 \), ensure the curve opens downward. Always remember that sketching isn't about precision; it's about understanding the behavior and features of the function visually. This skill helps you in interpreting and analyzing quadratic functions quicker and more effectively.