Problem 49 - Precalculus : Building Concepts and Connections

Question

Find the vertex and axis of symmetry of the associated parabola for each quadratic function.Then find at least two additional points on the parabola and sketch the parabola by hand. $$f(t)=\frac{1}{3}-3 t+t^{2}$$

Step-by-Step Solution

Answer

The vertex of the parabola is (1.5, -3.45), the axis of symmetry is \( t = 1.5 \), and two additional points on the parabola are (0, 1/3) and (3, 1/3). To complete the sketch, plot these points and the axis of symmetry.

Step 1: Find the Vertex

To find the vertex of the quadratic function, we use the formula \(-b/2a\). From the given quadratic function \(f(t)=t^{2}-3t+\frac{1}{3}\), \(a = 1\) and \(b = -3\). Hence, \(t = -(-3)/(2*1) = 1.5 \). To find the vertex, we substitute \(t = 1.5\) into the function: \( f(1.5) = (1.5)^{2}-3*1.5+\frac{1}{3}=0.75-4.5+\frac{1}{3}=-3.45 \). Hence, the vertex of the parabola is (1.5, -3.45)

Step 2: Find the Axis of Symmetry

The axis of symmetry is simply the x-coordinate of the vertex, which is \( t = 1.5 \).

Step 3: Find Additional Points

We choose two additional points by substituting different values for \( t \). Let's take \( t = 0 \) and \( t = 3 \). For \( t = 0\), we have \(f(0)=\frac{1}{3}\), and for \( t = 3 \), we have \(f(3)=(3)^{2}-3*3+\frac{1}{3}=9-9+\frac{1}{3}=\frac{1}{3}\). Hence, the two additional points are (0, 1/3) and (3, 1/3).

Step 4: Sketch the Parabola

Now that we have the vertex and three additional points, we can sketch the parabola by hand. The axis of symmetry will guide us to mirror our points.

Key Concepts

Vertex of a Quadratic Function

Understanding the vertex of a quadratic function unlocks the door to visualizing its graph, known as a parabola. The vertex is essentially the peak or the lowest point of the parabola, depending on whether the parabola opens upwards or downwards. This point is vital as it gives us a reference for the rest of the graph.

To find the vertex, the formula \( -b/(2a) \) is used, where \( a \) and \( b \) are coefficients from the function's standard form \( ax^2 + bx + c \). In our exercise, the function \( f(t) = t^2 - 3t + \frac{1}{3} \) gives us \( a = 1 \) and \( b = -3 \) leading to the vertex's \( t \) value of 1.5. Plugging \( t = 1.5 \) back into the function yields the \( y \) value, resulting in the vertex \( (1.5, -3.45) \).

Finding the vertex is a powerful first step in graphing a quadratic function and offers a benchmark for symmetry and orientation of the parabola.

Axis of Symmetry

The axis of symmetry is a fundamental concept when dealing with quadratic functions. It is a vertical line that runs through the vertex and divides the parabola into two mirror-image halves. Every parabola has one and only one axis of symmetry which can be used to predict other points on the graph.

The equation of the axis of symmetry can easily be found using the formula \( x = -b/(2a) \) which yields the same \( x \) value as that of the vertex. From our previous calculation for the vertex, \( t = 1.5 \) also describes the axis of symmetry for the parabola \( f(t) \). This helps ensure that points found on one side of the graph can be reflected across the axis to find corresponding points on the other side, thus shaping the parabola.

Graphing Parabolas

Graphing parabolas is a graphical representation of quadratic functions. The process involves plotting the vertex, the axis of symmetry, and additional points to reveal the distinct 'U' shaped curve. With the vertex as the starting point, you use the axis of symmetry to help find and plot additional points, ensuring that the graph is a mirrored image on either side.

By picking values of \( t \) and substituting them into the function, like \( t = 0 \) and \( t = 3 \) in our exercise, we obtain other points on the parabola which help in sketching a more accurate curve. It is especially beneficial to choose values of \( t \) that are equidistant from the axis of symmetry, as this ensures that our plotted points lead to a balanced graph which can be done smoothly by hand.

Quadratic Equation

The quadratic equation is at the heart of understanding quadratic functions and their properties. It takes the general form \( ax^2 + bx + c = 0 \), where \( a \) cannot be zero. Quadratic equations describe the relationship between the variable \( x \) and how it affects the resulting value of the equation, which we graph as a parabola.

A quadratic equation doesn't just help us find the roots or zeros of a function; it also helps in determining the vertex, the axis of symmetry and the shape of the graph. In the given exercise, \( f(t) = \frac{1}{3} - 3t + t^2 \) is in the form \( at^2 + bt + c \) from which we can extract values to find the vertex and axis of symmetry, thereby sketching the entire graph. Understanding the fact that all these features are interconnected could provide students a more holistic understanding of graphing parabolas.