Problem 32 - Precalculus : Building Concepts and Connections

Question

Use transformations to graph the quadratic function and find the vertex of the associated parabola. $$g(t)=2(t-3)^{2}+3$$

Step-by-Step Solution

Answer

The parabola opens upwards with vertex at point (3,3).

Step 1: Identify the Vertex

The vertex form of a parabola is \(g(t)=a(t-h)^{2}+k\), where \((h, k)\) are the coordinates of the vertex. Given the function \(g(t)=2(t-3)^{2}+3\), we can clearly see that the vertex of the parabola is at \((h,k)=(3,3)\).

Step 2: Identify the Direction

The coefficient a in the vertex form determines the direction of the parabola. If a is positive, the parabola opens upwards, and if a is negative, it opens downwards. For our function, a=2 which is positive, so the parabola will open upwards.

Step 3: Plot the Function

Now we can plot the function. Start by plotting the vertex at point (3,3). Since our parabola opens upwards and is stretched by a factor of 2, draw a parabola starting from this vertex point that opens upwards and is narrower than the standard parabola \(y=x^2\). The curve should also contain the points (4,5) and (2,5), which are derived from the standard movements in a parabola.

Key Concepts

Parabola

A parabola is a U-shaped curve that can open upwards or downwards on a graph. The general equation for a parabola in vertex form is \(y = a(x-h)^2 + k\). Changes to the value of \(a\) determine the direction and the width of the parabola. If \(a\) is positive, the parabola opens upwards. If \(a\) is negative, it opens downwards. Additionally, the larger the absolute value of \(a\), the narrower the parabola becomes. Conversely, a smaller absolute value means the parabola will be wider. Understanding the shape and direction of a parabola is crucial for graphing quadratic functions. These properties help you visualize the path that the parabola will take on a coordinate plane, which is why it is often taught in the context of quadratic functions.

Vertex

The vertex of a parabola is a significant point as it represents the extreme point on the curve. This means it's either the highest or lowest point on the graph, depending on the direction the parabola opens. In the vertex form \(y = a(x-h)^2 + k\), the vertex is found at the point \((h, k)\). For the function \(g(t)=2(t-3)^{2}+3\), the vertex is \((3,3)\). Knowing the vertex is important because it provides a reference point around which the entire parabola is shaped. This point helps in quickly sketching the parabola as it is the central point from which all points equidistant on the horizontal plane will lie vertically on either side.

Transformations

Transformations of a quadratic function involve shifting, stretching or compressing the parabola. The function \(g(t)=2(t-3)^{2}+3\) is a transformation of the simpler function \(y=x^2\). Transformations can include:

• Horizontal Shifts: In the term \((t-3)\), the \(-3\) indicates a shift 3 units to the right.
• Vertical Shifts: The \(+3\) moves the entire graph 3 units up.
• Vertical Stretch/Compression: The coefficient \(a=2\) causes a vertical stretch, making the parabola narrower compared to \(y=x^2\).

Transformations allow us to move and resize the parabola on a graph without altering its fundamental shape. Recognizing these changes in the equation helps to quickly identify how the graph will appear.

Graphing

Graphing a quadratic function helps visualize the parabola and understand its properties. For the function \(g(t)=2(t-3)^{2}+3\), we start by plotting its vertex at the point \((3,3)\). This point establishes the direction and position of the parabola on the graph. Because the coefficient \(a=2\) is positive, the parabola opens upwards. It is also narrower because of the vertical stretch factor given by \(a=2\). After plotting the vertex, additional points like \((4,5)\) and \((2,5)\) are derived from the standard movements and confirm the shape of the parabola. Graphing allows us to see how the mathematics of quadratic functions translate visually, offering a clearer understanding of how changes in the equation affect the graph.