Question

a) Describe the relationship between the graphs of \(y=x^{2}\) and \(y=3(x-4)^{2}+2\) b) Predict the relationship between the graphs of \(y=x^{4}\) and \(y=3(x-4)^{4}+2\) c) Verify the accuracy of your prediction in part b) by graphing using technology.

Short answer

The graph of \(y=3(x-4)^2 + 2\) is a right shift by 4 units, stretched by 3 and shifted up by 2 units from \(y=x^2\). Similar transformations apply predictably from \(y=x^4\) to \(y=3(x-4)^4 + 2\), confirming the transformation method's consistency.

Step-by-step solution

Introduction to Quadratic Transformations

Consider the basic quadratic function graph, which is the graph of the function \(y = x^2\). This graph is a parabola that opens upwards with its vertex at the origin (0,0).

Analyze \(y=3(x-4)^2 + 2\)

To describe the relationship between \(y = x^2\) and \(y = 3(x-4)^2 + 2\), note the transformations: (1) The \((x-4)\) indicates a horizontal shift to the right by 4 units. Thus, the vertex moves from the origin to (4,0). (2) The coefficient 3 indicates a vertical stretch by a factor of 3. (3) The +2 translates the graph 2 units upwards. The vertex is now at (4,2).

Prediction for Quartic Function Transformations

Considering the transformation pattern: For \(y=x^4\) to \(y=3(x-4)^4+2\), similar horizontal (right by 4 units), vertical stretch (by factor 3), and vertical shift (up by 2 units) transformations are expected. The graph will vertically stretch by a factor of 3, shift 4 units to the right, and 2 units upward. The vertex for \(y=3(x-4)^4+2\) would be at (4,2).

Verify Using Graphing Technology

Use a graphing calculator or graphing software to plot both functions \(y = x^4\) and \(y = 3(x-4)^4 + 2\). Confirm if the transformations similarly adjust the graph of the quartic function verifying the above shifts, stretches and new vertex match the prediction. Observing both graphs should illustrate transformations occurring correctly as described.

Key concepts

Quadratic Function

The quadratic function, represented as \(y = x^2\), is one of the most fundamental functions in mathematics. Its graph forms a parabola that opens upwards and has its vertex at the origin (0,0). This symmetrical shape makes it easily recognizable. Understanding this basic form is crucial because many transformations are applied to it in more complex functions.

Quartic Function

A quartic function is represented in the form of \(y = x^4\). Graphs of quartic functions are generally 'W-shaped' or 'M-shaped,' depending on the leading coefficient's sign. Similar to quadratic functions, quartic functions can be transformed by applying various changes. The general transformation rules apply here as well, involving stretches, shifts, and translations.

Graph Transformations

Graph transformations involve changing the position or shape of the graph of a function. These transformations include:

• Translation: Moving the graph without changing its shape or orientation.
• Stretching: Changing the graph's shape by stretching vertically or horizontally.
• Reflection: Flipping the graph over a specific axis.

Understanding these transformations helps in predicting how alterations in a function's equation will affect its graph.

Vertical Stretch

A vertical stretch involves multiplying the function by a coefficient greater than 1 outside of the function. For instance, in the function \(y = 3(x-4)^2 + 2\), the coefficient 3 causes a vertical stretch by a factor of 3. This means that each y-value is multiplied by 3, making the graph appear taller and narrower compared to the original graph of \(y = (x-4)^2 + 2\).

Horizontal Shift

Horizontal shifts occur when the function’s input (x) is adjusted. For example, in \(y = 3(x-4)^2 + 2\), the term \(x-4\) indicates a horizontal shift 4 units to the right. This shift can be understood as moving every point of the graph 4 units rightward. Understanding horizontal shifts is essential as they directly alter the position of the graph's vertex along the x-axis.

Vertical Shift

Vertical shifts involve adding or subtracting a constant to/from the function. In the function \(y = 3(x-4)^2 + 2\), the +2 translates the graph 2 units upward. It modifies the y-values by increasing each one by 2. This results in the entire graph moving up without changing its shape. Knowing vertical shifts helps in accurately placing the graph along the y-axis.