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Chapter 6: Problem 17

State the rule that describes the function \(y=3\left(x^{2}-1\right)\)

Short Answer

Expert verified
The function is a vertically stretched and downward shifted quadratic: \( y = 3x^2 - 3 \).

Step by step solution

01

Identify the Type of Function

The equation given is \( y = 3(x^2 - 1) \). This is a quadratic function, which is typically written in the form \( y = ax^2 + bx + c \). In this example, \( a = 3 \), \( b = 0 \), and \( c = -3 \).
02

Simplify the Function

Expand the expression by distributing the 3: \[ y = 3(x^2) - 3 \times 1 = 3x^2 - 3 \]. The function can now be seen as \( y = 3x^2 - 3 \), clearly showing it is a quadratic function.
03

Determine the Transformation Effects

The function is a transformation of the basic quadratic function \( y = x^2 \). Multiplying by 3 vertically stretches the parabola by a factor of 3, and subtracting 3 shifts the parabola down by 3 units on the y-axis.
04

State the Function Rule

Based on the transformations, the rule describing the function is that it is a vertically stretched and shifted downward quadratic function, \( y = 3x^2 - 3 \).

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Key Concepts

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

Function Transformation
Function transformation refers to modifying the basic form of a function to create a new function. Transformations can include shifting, reflecting, stretching, or compressing a function's graph.
In the context of quadratic functions, like the one given, transformations are applied to the basic quadratic function, typically written as \( y = x^2 \). This exercise involves two primary transformations:
  • **Vertical Stretch:** When a function is multiplied by a coefficient (other than one), like the 3 in \( y = 3(x^2 - 1) \), the graph of the parabola is stretched vertically. This means it becomes narrower compared to the basic \( y = x^2 \) parabola.
  • **Vertical Shift:** Adding or subtracting a number from the function results in a shift along the y-axis. Here, subtracting 3 (-3) from \( x^2 \) moves the entire graph downward by 3 units.
These transformations allow us to manipulate the visual presentation of a function's graph, which helps in analyzing and understanding its behavior beyond its basic form.
Parabola
A parabola is the U-shaped graph that represents a quadratic function. It is defined by its symmetry and can either open upwards or downwards depending on the sign of the leading coefficient in the quadratic equation.
The equation \( y = 3x^2 - 3 \) represents a parabola that opens upwards because the leading coefficient, 3, is positive.
Key features of a parabola include its vertex, axis of symmetry, and direction of opening:
  • **Vertex:** This is the highest or lowest point on the parabola. For an upward opening parabola, it is the lowest point.
  • **Axis of Symmetry:** This is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.
  • **Direction of Opening:** Determined by the sign of the leading coefficient; positive coefficients result in an upward-opening parabola, whereas negative ones cause a downward opening.
Understanding these properties helps in sketching the graph and predicting the behavior of the quadratic function under various transformations.
Vertex
The vertex of a parabola is a crucial concept in understanding quadratic functions. It represents the point of minimum or maximum value of the function.
For the quadratic function given by \( y = 3x^2 - 3 \), the vertex can be found using the formula for the vertex of a parabola:
  • **Formula:** The vertex of a quadratic function \( y = ax^2 + bx + c \) is given by the point \( (h, k) \). In the case of this simplified form \( y = ax^2 + bx + c \), the horizontal position \( h \) can be calculated using \( h = -\frac{b}{2a} \).
  • **Example Calculation:** Here, \( b = 0 \) and \( a = 3 \), so \( h = -\frac{0}{2 imes 3} = 0 \). The vertex coordinates are \( (0, -3) \).
The vertex is crucial as it provides a starting point for graphing the parabola and gives insight into the function's maximum or minimum values, offering a deeper understanding of its real-life applications.

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Most popular questions from this chapter

The ramp function is defined by $$ f(x)= \begin{cases}0 & x<0 \\ k x & x \geq 0\end{cases} $$ (a) Sketch a graph of this function. (b) State the position of any discontinuities. (c) Find \(\lim _{x \rightarrow 0} f(x)\).

If \(f(x)=5-4 x\) find \(f^{-1}(x)\). Show that \(f\left(f^{-1}(x)\right)=x\)

The point \(\mathrm{A}\left(x_{1}, y_{1}\right)\) lies on the line \(y=-2 x+3 .\) If the value of \(x_{1}\) is increased by 7, what is the resulting change in the value of \(y_{1}\) ?

(a) Sketch a graph of the function \(u(t-1)-u(t-2)\) (b) State the position of any discontinuities.

Write a formula for the function given by the rule 'subtract the cube of the input from the square of the input'.
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