Question

Use a graphing calculator to graph the function and its parent function. Then describe the transformations. \(f(x)=4 x^2-3\)

Short answer

The function \(f(x)=4x^2-3\) is the result of the parent function \(y=x^2\) undergoing a vertical stretch of factor 4 and a vertical shift downwards by 3 units.

Step-by-step solution

Identify the parent function

The parent function in this case will be the basic quadratic function \(y=x^2\).

Graph the parent function

Plot the function \(y=x^2\) on the graph. This will give a parabolic curve opening upwards.

Identify the transformation

Looking at the given function \(f(x)=4x^2-3\), two transformations are clear. A vertical stretch of factor 4 i.e. the graph will be narrower because of \(4x^2\) (where \(4>1\)). And a vertical shift of 3 units downwards because of \(-3\).

Graph the transformed function

Plot the function \(f(x)=4x^2-3\) which includes the transformation on the parent function graph. The curve will be more steep and 3 units lower compared to the parent graph of \(y=x^2\).

Key concepts

Parent Function

The term 'parent function' refers to the simplest form of a function family, which serves as the foundation for understanding more complex forms of that function. In the case of quadratic functions, the parent function is represented by the equation y = x^2. This is the most basic form of a quadratic equation, and it produces a U-shaped graph known as a parabola.

When graphing the parent function y = x^2, we observe that it is symmetric about the y-axis and has its vertex at the origin (0,0). The parabola opens upwards since the coefficient of x^2 is positive. Understanding the parent function is crucial because any transformation of the quadratic function can be seen as an alteration from this original state, making it easier for students to visualize and predict how different coefficients and constants will affect the graph's shape and position.

Quadratic Transformations

Quadratic transformations involve modifying the parent function in various ways to change the appearance of the graph. These modifications include vertical stretches/shrinks, reflections, horizontal shifts, and vertical shifts. In the example f(x) = 4x^2 - 3, we see two key transformations:

• A vertical stretch by a factor of 4, which occurs due to the multiplication of x^2 by 4. This makes the graph narrower or steeper than that of the parent function.
• A vertical shift downward by 3 units, indicated by the subtraction of 3 from 4x^2. This moves the entire graph of the parent function down so that its vertex is no longer at (0,0), but at (0, -3).

Understanding how these transformations affect the graph helps students become adept at not just graphing quadratic functions, but also interpreting the significance of different coefficients and constants within the function's formula.

Graphing Calculator

A graphing calculator is a powerful educational tool that assists students in visualizing equations and their corresponding graphs. To graph the function f(x) = 4x^2 - 3 along with its parent function, students can use a graphing calculator to plot both equations and observe the transformations directly.

Using the graphing calculator, one can enter the equation of the parent function and then the transformed function to see the effects of the transformations. This visual aid is particularly helpful for understanding abstract concepts like stretching and shifting. Moreover, graphing calculators often have features that allow for tracing values along the curve, finding intercepts, and analyzing the vertex, which furthers a student's comprehension of quadratic functions and their graphs. It is essential, however, that students first understand the underlying concepts before relying on technology to ensure a deep and lasting grasp of the material.