Problem 42
Question
Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$x=2 x^{2}-y+4$$
Step-by-Step Solution
Verified Answer
The equation represents a parabola with a vertical axis.
1Step 1: Rearrange the Equation
The given equation is \( x = 2x^2 - y + 4 \). Let's rearrange it to express \( y \) in terms of \( x \), leading to \( y = 2x^2 - x + 4 \).
2Step 2: Identify the Conic Section Type
The equation \( y = 2x^2 - x + 4 \) can be recognized as a quadratic equation in \( x \,or\, y\). This fits the standard form \( y = ax^2 + bx + c \), which represents a parabola.
3Step 3: Determine the Axis Orientation
Since the equation is in the form \( y = ax^2 + bx + c \), which represents a parabolic equation where \( x \) is squared and not \( y \), this parabola has a vertical axis of symmetry.
Key Concepts
ParabolaVertical AxisQuadratic EquationGraph Identification
Parabola
When we think of conic sections, one of the most common structures that come to mind is the parabola.
Parabolas have unique characteristics that make them stand out from other types of curves. They are symmetrical U-shaped figures that open in either a vertical or horizontal direction, depending on their orientation.
A parabola is embodied by its standard quadratic form, which reveals its key features such as its vertex, axis of symmetry, and direction of opening. This helps us in identifying and graphing parabolas with accuracy. Understanding a parabolic equation in its proper form makes it easier to manipulate and analyse within mathematical problems.
Parabolas have unique characteristics that make them stand out from other types of curves. They are symmetrical U-shaped figures that open in either a vertical or horizontal direction, depending on their orientation.
A parabola is embodied by its standard quadratic form, which reveals its key features such as its vertex, axis of symmetry, and direction of opening. This helps us in identifying and graphing parabolas with accuracy. Understanding a parabolic equation in its proper form makes it easier to manipulate and analyse within mathematical problems.
Vertical Axis
In mathematics, identifying the axis of a parabola is crucial. It determines the direction in which the parabola opens.
For parabolas described by equations of the form \(y = ax^2 + bx + c\), the axis of symmetry is vertical. This means the parabola opens either upwards or downwards.
Conversely, if the parabola equation were in the form \(x = ay^2 + by + c\), the opening would be horizontal, and thus, the axis of symmetry would be horizontal as well. Knowing the axis orientation helps in plotting the parabola correctly and understanding how it interacts with other mathematical elements in its vicinity.
For parabolas described by equations of the form \(y = ax^2 + bx + c\), the axis of symmetry is vertical. This means the parabola opens either upwards or downwards.
Conversely, if the parabola equation were in the form \(x = ay^2 + by + c\), the opening would be horizontal, and thus, the axis of symmetry would be horizontal as well. Knowing the axis orientation helps in plotting the parabola correctly and understanding how it interacts with other mathematical elements in its vicinity.
Quadratic Equation
A quadratic equation is central to describing parabolas. It usually takes the form \(ax^2 + bx + c = 0\). When a quadratic equation is expressed as \(y = ax^2 + bx + c\), it's easy to observe how it translates into a parabolic curve.
This equation allows us to recognize the parabola's key elements, such as its concavity (which direction it opens) based on the sign of \(a\), and the location of its vertex and roots.
The quadratic nature indicates that there is only one squared term involved, thereby narrowing the shape to a parabola specifically. Mastery of quadratic equations enables effective graphing and manipulation of parabolas in different mathematical contexts.
This equation allows us to recognize the parabola's key elements, such as its concavity (which direction it opens) based on the sign of \(a\), and the location of its vertex and roots.
The quadratic nature indicates that there is only one squared term involved, thereby narrowing the shape to a parabola specifically. Mastery of quadratic equations enables effective graphing and manipulation of parabolas in different mathematical contexts.
Graph Identification
Being able to identify the kind of graph represented by a conic equation is a vital skill in mathematics.
In our case, we identified the given equation \(y = 2x^2 - x + 4\) as a parabola due to its quadratic form that features a single squared variable.
Understanding the specific form a graph takes helps predict its shape and behavior on a coordinate plane. Recognizing whether an equation represents a parabola, circle, ellipse, or hyperbola leads directly to understanding that graph's properties, such as symmetry, vertices, foci, and directrices.
Graph identification thus forms the foundational block for solving and interpreting broader mathematical problems efficiently.
In our case, we identified the given equation \(y = 2x^2 - x + 4\) as a parabola due to its quadratic form that features a single squared variable.
Understanding the specific form a graph takes helps predict its shape and behavior on a coordinate plane. Recognizing whether an equation represents a parabola, circle, ellipse, or hyperbola leads directly to understanding that graph's properties, such as symmetry, vertices, foci, and directrices.
Graph identification thus forms the foundational block for solving and interpreting broader mathematical problems efficiently.
Other exercises in this chapter
Problem 41
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