Question

Graph the function \(f\) by starting with the graph of \(y=x^{2}\) and using transformations. \(f(x)=\frac{2}{3} x^{2}+\frac{4}{3} x-1\)

Short answer

The graph of \( f(x) = \frac{2}{3} x^2 + \frac{4}{3} x - 1 \) is a vertically compressed parabola with vertex at \((-1, -\frac{5}{3})\), opening upwards and being wider than the standard parabola \( y = x^2 \).

Step-by-step solution

Identify the base function

The base function is given as: \[ y = x^2 \]This is a standard parabola opening upwards, centered at the origin (0, 0).

Recognize the transformations

The function to be graphed is:\[ f(x) = \frac{2}{3}x^2 + \frac{4}{3}x - 1 \]Recognize the form of the quadratic function:\[ f(x) = ax^2 + bx + c \]Here, \(a = \frac{2}{3}\), \(b = \frac{4}{3}\), and \(c = -1\).

Apply the vertical stretch

The coefficient of \(x^2\) is \(\frac{2}{3}\), which means the parabola has been vertically compressed by a factor of \(\frac{2}{3}\) compared to the standard parabola \( y = x^2 \). Therefore, the parabola will be wider.

Complete the square

Rewriting the function in vertex form by completing the square:\[ f(x) = \frac{2}{3}x^2 + \frac{4}{3}x - 1 \]First, factor out \(\frac{2}{3}\) from the quadratic and linear terms:\[ f(x) = \frac{2}{3}(x^2 + 2x) - 1 \]Complete the square inside the parentheses:\[ x^2 + 2x = (x + 1)^2 - 1 \]Thus, the function becomes:\[ f(x) = \frac{2}{3}((x + 1)^2 - 1) - 1 \]Simplify the expression:\[ f(x) = \frac{2}{3}(x + 1)^2 - \frac{2}{3} - 1 \]Combine the constants:\[ f(x) = \frac{2}{3}(x + 1)^2 - \frac{5}{3} \]So the vertex form of the function is:\[ f(x) = \frac{2}{3}(x + 1)^2 - \frac{5}{3} \]

Identify the vertex and axis of symmetry

From the vertex form, \( f(x) = \frac{2}{3}(x + 1)^2 - \frac{5}{3} \), identify the vertex:Vertex: \((-1, -\frac{5}{3})\)The axis of symmetry is the vertical line through the vertex, which is:\[ x = -1 \]

Sketch the transformed graph

To sketch the graph of \( f(x) = \frac{2}{3} x^2 + \frac{4}{3} x - 1 \), follow these steps:1. Plot the vertex at \((-1, -\frac{5}{3})\).2. Draw the axis of symmetry at \(x = -1\).3. Since the parabola is vertically compressed by a factor of \(\frac{2}{3}\), it opens upwards and is wider than the standard parabola \( y = x^2 \).4. Plot additional points around the vertex to help shape the parabola.

Key concepts

transformations

In mathematics, transformations help us understand how a function changes when it is shifted, stretched, or reflected. For the quadratic function, consider the base function as \(y = x^2\). When graphing \(f(x) = \frac{2}{3}x^2 + \frac{4}{3}x - 1\), we need to apply several transformations to the base function:

• Vertical Stretch/Compression: The coefficient of \(x^2\) is \(a\). If \( |a| < 1\), the graph is vertically compressed; otherwise, if \( |a| > 1\), it is vertically stretched. Here, \(a = \frac{2}{3}\), so the parabola is vertically compressed by a factor of \( \frac{2}{3}\), making it wider.
• Horizontal Shifts: Shifting the function left or right is often easier seen in vertex form.
• Vertical Shifts: The constant term \(c = -1\) shifts the graph downward.

These transformations give us the new shape and position of the parabola.

vertex form

The vertex form of a quadratic function provides valuable information for graphing. Vertex form is given as: \[ f(x) = a(x - h)^2 + k \]Here, \( (h, k)\) is the vertex of the parabola, and \(a\) affects the width and direction. Converting from standard form \( f(x) = \frac{2}{3}x^2 + \frac{4}{3}x - 1 \) to vertex form involves completing the square. This form makes it easy to identify the vertex and other characteristics of the parabola. For our function, we have: \[ f(x) = \frac{2}{3}(x + 1)^2 - \frac{5}{3} \]Here, the vertex is \( (-1, -\frac{5}{3})\). This tells us the highest or lowest point, the direction of opening, and helps in sketching the graph accurately.

completing the square

Completing the square is a method used to convert a quadratic function from standard form to vertex form. Here's how it's done step-by-step for \( f(x) = \frac{2}{3}x^2 + \frac{4}{3}x - 1 \):
1. Factor out the leading coefficient from the quadratic and linear terms: \[ f(x) = \frac{2}{3}(x^2 + 2x) - 1 \]
2. Inside the parentheses, complete the square: \[ x^2 + 2x = (x + 1)^2 - 1 \]
3. Substitute back and simplify: \[ f(x) = \frac{2}{3}((x + 1)^2 - 1) - 1 \] \[ f(x) = \frac{2}{3}(x + 1)^2 - \frac{2}{3} - 1 \] \[ f(x) = \frac{2}{3}(x + 1)^2 - \frac{5}{3} \]
This process helps us see transformations more clearly and identify the vertex.

parabolas

Parabolas are the graphs of quadratic functions and have a distinctive U-shape. The key features of parabolas include the vertex, axis of symmetry, and direction of opening. For \( f(x) = \frac{2}{3}x^2 + \frac{4}{3}x - 1 \):

• Vertex: The vertex is the point \( (-1, -\frac{5}{3})\), which is the parabola's lowest point since it opens upwards.
• Axis of Symmetry: The vertical line passing through the vertex. For this parabola, it is \( x = -1 \).
• Direction of Opening: The parabola opens upward because \(a = \frac{2}{3} > 0\).
• Width: This parabola is wider than the standard \[y = x^2\] due to the vertical compression factor \( \frac{2}{3} \).

Understanding these features helps us graph them accurately and recognize the effect of different coefficients.