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Chapter 1: Problem 21

Graph the function and its parent function. Then describe the transformation. \(f(x)=2 x^2\)

Short Answer

Expert verified
The given function \(f(x)=2 x^2\) is a vertical stretch of its parent function \(g(x) = x^2\) by a factor of 2. The graph of \(f(x)=2 x^2\) is narrower compared to \(g(x) = x^2\).

Step by step solution

01

Identify the Parent Function

Start by identifying the parent function. In this case, our parent function is \(g(x) = x^2\). This is a basic quadratic function, which looks like a U-shape or a parabola.
02

Understand Transformations

The next step is to understand what the given function \(f(x)=2 x^2\) tells us about transformations. The 2 in this function 'stretches' it vertically by a factor of 2.
03

Graph the Parent Function

Using graph paper or graphing software, graph the parent function \(g(x) = x^2\). Note that it passes through the point (0,0) - the origin, and the curve is symmetrical about the y-axis.
04

Graph the Given Function

Next, graph the given function \(f(x)=2 x^2\).Since there's a vertical stretch, the curve of the graph will be narrower than that of the parent function.
05

Describe Transformations

Finally, describe the transformation from \(g(x) = x^2\) to \(f(x)=2 x^2\). The function \(f(x)=2 x^2\) is a vertical stretch of \(g(x) = x^2\) by a factor of 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parabola
A parabola is a U-shaped curve that is the graph of a quadratic function. Quadratic functions are essential in algebra and take the form \( y = ax^2 + bx + c \). The parabolas can open upwards or downwards; in our example, with \( f(x) = 2x^2\), it opens upwards because the coefficient of \( x^2 \), which is 2, is positive.
  • An upward-opening parabola has a lowest point known as the vertex. In the parent function \( g(x) = x^2 \), the vertex is at the origin (0,0).
  • The y-axis is the line of symmetry. This means if you fold the parabola along the y-axis, both halves will match perfectly.
  • The wider the parabola, the smaller the coefficient of \( x^2 \).
  • The more narrow the parabola, the larger the coefficient of \( x^2 \).
Understanding how parabolas behave makes it easier to graph and manipulate quadratic functions.
Parent Function
The parent function for all quadratic functions is the simplest quadratic equation, represented by \( g(x) = x^2 \). This function forms the basis for many transformations and is crucial in understanding more complex quadratic functions.
  • It forms a standard parabola that opens upwards with a vertex at the origin.
  • The parent function is symmetrical about the y-axis and shows how every quadratic function evolves through modifications.
  • It passes through points such as (0,0), (1,1), and (-1,1), which helps in plotting its graph accurately.
By understanding the parent function, students can easily transition into learning about how changes in its equation affect its graph.
Transformation
Transformations involve altering the parent function to create a new graph through actions such as stretching, shrinking, translating, or reflecting. In the case of \( f(x) = 2x^2 \), we have a vertical stretch.
  • The coefficient 2 in \( f(x) = 2x^2 \) stretches the parabola vertically by a factor of 2, making it narrower than the parent function.
  • Vertical stretches affect the y-values by multiplying them, moving points away from the x-axis.
Overall, understanding transformations is like mastering a toolkit that modifies the shape and position of the parabolas on the graph.
Graphing
Graphing a quadratic function involves plotting points and understanding its movements via transformations. Let's focus on sketching \( f(x) = 2x^2 \) and its parent function.
  • Begin by graphing the parent function \( g(x) = x^2 \). Plot key points like (0,0), (1,1), and (-1,1).
  • Once the parent parabola is drawn, apply the vertical stretch to all y-values to graph \( f(x) = 2x^2 \). This means doubling each y-value; thus, the graph appears narrower or steeper.
  • Ensure symmetry is maintained about the y-axis. The vertex remains in the same position unless other different transformations are applied.
Graphing provides visual insights into how functions behave and aids in deeper comprehension of quadratic equations.

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