Problem 88
Question
I represented \(y=x^{2}-9\) with the parametric equations \(x=t^{2}\) and \(y=t^{4}-9\).
Step-by-Step Solution
Verified Answer
The correct parametric equations for \(y=x^{2}-9\) would be \(x=t\) and \(y=t^{2}-9\). The proposed parametric equations \(x=t^{2}\) and \(y=t^{4}-9\) do not represent \(y=x^{2}-9\).
1Step 1: Examine the original function
The original function \(y=x^{2}-9\) is a standard upward-facing parabola shifted down by 9 units.
2Step 2: Identify the issue in the parametric representation
For correct parametric equations, squaring \(x=t^{2}\) should give \(x^{2}=t^{4}\). However, this would mean that \(y=t^{4}\) for the function \(y=x^{2}\), which is not the case in the function provided \(y=t^{4}-9\).
3Step 3: Access and Propose a Fix
Keeping our original function \(y=x^{2}-9\) the correct parametric representation would be \(x=t\) and \(y=t^{2}-9\), not \(x=t^{2}\) and \(y=t^{4}-9\). This correctly depicts the original function.
4Step 4: Confirm the Fix
To confirm this, if we substitute \(t=x\) from the first equation into the second equation we will get \(y=x^{2}-9\) which matches the original function.
Key Concepts
Understanding the ParabolaParametric Representation ExplainedFunction TransformationNavigating Coordinate Systems
Understanding the Parabola
A parabola is a symmetrical curve that is shaped like an open bowl. This distinctive shape occurs in many aspects of our daily lives, like car headlights and the paths of basketball shots. Mathematically, a parabola can be represented as a quadratic function, such as the straightforward equation \( y = x^2 \), which opens upwards.
In the given exercise, the function \( y = x^2 - 9 \) describes a parabola that has been shifted down by 9 units. This vertical shift doesn't affect the shape of the parabola itself – it just adjusts its position along the y-axis. Thus, the vertex of this parabola, originally at \( (0,0) \) in the case of \( y = x^2 \), is moved down to \( (0,-9) \).
When analyzing a parabola, key features to take note of include the vertex, axis of symmetry, and whether it opens upwards or downwards. These characteristics are essential in both standard equations and when using parametric equations.
In the given exercise, the function \( y = x^2 - 9 \) describes a parabola that has been shifted down by 9 units. This vertical shift doesn't affect the shape of the parabola itself – it just adjusts its position along the y-axis. Thus, the vertex of this parabola, originally at \( (0,0) \) in the case of \( y = x^2 \), is moved down to \( (0,-9) \).
When analyzing a parabola, key features to take note of include the vertex, axis of symmetry, and whether it opens upwards or downwards. These characteristics are essential in both standard equations and when using parametric equations.
Parametric Representation Explained
Parametric representation offers a different approach to describe curves. Instead of expressing \( y \) solely in terms of \( x \), we introduce a third variable, typically known as the parameter \( t \). This allows us to represent both coordinates of a point as functions of \( t \), such as \( x = f(t) \) and \( y = g(t) \).
In essence, the parameter \( t \) 'guides' us along the curve, specifying a set of coordinates for each value of \( t \). This representation can be especially useful for tracing complex trajectories or when dealing with motion along a path.
In the context of our parabola \( y = x^2 - 9 \), a correct parametric representation would involve choosing expressions for \( x \) and \( y \) that, when combined, recover the original equation. An effective choice is \( x = t \) and \( y = t^2 - 9 \), where a scan through different values of \( t \) will recreate the parabola's precise form in the \( xy \)-plane.
In essence, the parameter \( t \) 'guides' us along the curve, specifying a set of coordinates for each value of \( t \). This representation can be especially useful for tracing complex trajectories or when dealing with motion along a path.
In the context of our parabola \( y = x^2 - 9 \), a correct parametric representation would involve choosing expressions for \( x \) and \( y \) that, when combined, recover the original equation. An effective choice is \( x = t \) and \( y = t^2 - 9 \), where a scan through different values of \( t \) will recreate the parabola's precise form in the \( xy \)-plane.
Function Transformation
Function transformation involves modifying basic functions like \( y = x^2 \) through shifts, stretches, compressions, or reflections to generate new functions. Recognizing how these transformations work can reveal much about the resulting graph's behavior and appearance.
The function \( y = x^2 - 9 \) is a transformation of the basic parabola \( y = x^2 \), achieved by shifting it 9 units downward. Such transformations are crucial because they alter the graph's position, size, or orientation without changing the overall shape.
By understanding these transformations, we can predict changes in the graph by simply examining the algebraic expression.
The function \( y = x^2 - 9 \) is a transformation of the basic parabola \( y = x^2 \), achieved by shifting it 9 units downward. Such transformations are crucial because they alter the graph's position, size, or orientation without changing the overall shape.
- Vertical Shifts: Adding or subtracting a constant will move the function up or down.
- Horizontal Shifts: Replacing \( x \) with \( x \pm c \) will shift the graph sideways.
- Stretches/Compressions: Multiplying functions by constants will stretch or compress them.
- Reflections: Multiplying by negative values flips the graph across the axis.
By understanding these transformations, we can predict changes in the graph by simply examining the algebraic expression.
Navigating Coordinate Systems
Coordinate systems like the Cartesian coordinate plane help us represent and analyze geometric shapes analytically. The basic elements are the \( x \)-axis and \( y \)-axis, dividing the plane into four quadrants. Points in this two-dimensional plane are denoted as \( (x, y) \).
Additionally, when we use parametric equations, we often traverse these coordinate systems using a parameter \( t \) to describe the location on a curve. Each value of \( t \) finds a point \( (x, y) \) as specified by its parametric formula.
Understanding how to navigate these systems is crucial for graphing functions, particularly nonlinear ones like parabolas. By shifting, stretching, or rotating within the coordinate plane, we can visualize transformations and better comprehend the spatial nature of mathematical functions.
Additionally, when we use parametric equations, we often traverse these coordinate systems using a parameter \( t \) to describe the location on a curve. Each value of \( t \) finds a point \( (x, y) \) as specified by its parametric formula.
Understanding how to navigate these systems is crucial for graphing functions, particularly nonlinear ones like parabolas. By shifting, stretching, or rotating within the coordinate plane, we can visualize transformations and better comprehend the spatial nature of mathematical functions.
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