Question

Identify the function family to which \(f\) belongs. Compare the graph of \(f\) to the graph of its parent function. \(f(x)=-2 x^2+3\)

Short answer

The function \(f(x)=-2x^2+3\) belongs to the family of quadratic functions. Its parent function is \(f(x) = x^2\). The graph of the given function is a vertically reflected and upward shifted parabola as compared to the parent function.

Step-by-step solution

Identify the Function Family

Considering the function \(f(x)=-2 x^2+3\), this function is a quadratic function. The general form of a quadratic function is \(ax^2+bx+c\). In this case, \(a = -2\), \(b = 0\) and \(c = 3\). Hence, \(f\) belongs to the family of quadratic functions.

Identify the Parent function

The parent function of the quadratic family is \(f(x) = x^2\). Parent functions are the simplest form of the type of function being discussed. In this case, the most simple quadratic function is \(x^2\).

Compare the Graphs

The graph of the given function \(f(x)=-2 x^2+3\) and its parent function \(f(x)=x^2\) would be parabolas. However, the given function is a vertically reflected and shifted upwards form of the parent function because of the factors -2 and +3 respectively. The negative sign inverts the parent function and the +3 shifts it upwards by 3 units. These differences should be apparent when both the functions are graphed.

Key concepts

Function Families

In mathematics, understanding function families is essential when studying different types of functions. Function families are groups of functions that share common characteristics or behaviors, often described by a specific equation form. For example, one of the common function families is the family of quadratic functions. These functions are characterized by their highest degree of two, which means the variable is squared in their general form:

• The general expression for a quadratic function is: \( ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants.
• In the specific example given, \(f(x) = -2x^2 + 3\), it is clear that \(b = 0\), which simplifies the equation, but it still belongs to the quadratic family because the highest power of \(x\) is 2.
• Other function families include linear functions, exponential functions, and logarithmic functions, each defined by different characteristic equations.

Recognizing the family to which a function belongs aids in predicting its graph and behavior across different values of \(x\).
Function families simplify complex problems by grouping functions with shared characteristics, making them easier to analyze and understand.

Parent Function

The concept of a parent function is crucial when analyzing functions because it serves as the simplest form of the functions within a particular family. The parent function showcases the fundamental behavior of the function family without any transformations like shifts, stretches, or reflections.

• In the case of quadratic functions, the parent function is \( f(x) = x^2 \). This is because it has no additional terms that alter its basic shape.
• When considering transformations applied to a quadratic function, any variations from \( x^2 \) are due to these modifications, such as adding, subtracting, multiplying, or dividing by constants.

The parent function acts as a reference point. Understanding it allows students to see how modifications influence the shape and position of the graph.
For instance, comparing our example function \( f(x) = -2x^2 + 3 \) to its parent demonstrates how constants change graph features without losing the essential parabolic shape.

Graph Comparison

Comparing graphs is a vital skill that involves analyzing how different equations and their components influence a function's graph. When comparing a given function's graph to its parent function, key transformations include reflections, shifts, stretches, and compressions.

• In our example, the function \( f(x) = -2x^2 + 3 \) undergoes several transformations:
  • The leading negative sign (\(-2\)) indicates the parabola opens downward, a vertical reflection over the x-axis from its parent \( f(x) = x^2 \).
  • The coefficient \(-2\) causes a vertical stretch, making the parabola narrower compared to \( x^2 \).
  • The constant \(+3\) shifts the entire graph up by 3 units along the y-axis, altering its vertex from the origin \((0,0)\) to \((0,3)\).

These transformations show detailed alterations from the parent graph while retaining the basic structure of a parabola. Grasping these changes helps students predict behavior and shape based on function equations.
Essentially, understanding how graphs are transformed helps in decoding the equation itself and enables better insight for solving complex problems.