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Chapter 2: Problem 48

Write the standard form of the equation of the parabola that has the indicated vertex and passes through the given point. Vertex: (4,-1)\(;\) point: (2,3)

Short Answer

Expert verified
The standard form of the equation of the parabola is \(y = (x - 4)^2 - 1\).

Step by step solution

01

Identify the Vertex

Recognize the vertex (h,k) = (4,-1). The vertex form of a parabola is \(y = a(x - h)^2 + k\), so it is essential to know the vertex.
02

Substitute Vertex Details

Substitute the vertex into the parabola vertex format to have \(y = a(x - 4)^2 - 1\). The value 'a' is still unknown.
03

Use an Extra Point to Determine 'a'

Use another point on the parabola (x,y) = (2,3) to solve for 'a'. Substituting this point into the equation yields \(3 = a(2 - 4)^2 - 1\). This simplifies to \(3 = 4a - 1\), which further simplifies to \(a = 1\).
04

Substitute 'a' into the Equation

With the now found 'a' value, substitute it back into the parabola equation to yield \(y = (x - 4)^2 - 1\).

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Key Concepts

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

Vertex Form of a Parabola
The vertex form of a parabola gives you a nice, clear picture of how the parabola is positioned on a graph. It is written as: \[ y = a(x - h)^2 + k \] Here,
  • \( (h, k) \) is the vertex of the parabola, which is the turning point or the highest or lowest point, depending on its orientation.
  • The letter \( a \) determines how "wide" or "narrow" the parabola is and whether it opens upwards or downwards. A positive \( a \) indicates it opens upwards while a negative \( a \) means it opens downwards.
To express a parabola in vertex form, it is important to know the vertex. This form is particularly helpful because changing the values of \( h \) and \( k \) slides the parabola horizontally and vertically on the graph, respectively.
Standard Form of a Parabola
The standard form of a parabola gives another representation of the same curve as the vertex form, but appears as: \[ y = ax^2 + bx + c \] Here's what the different parts represent:
  • \( a \) again affects the direction and "width" of the parabola.
  • \( b \) and \( c \) control the position of the parabola on the x-y plane.
The standard form is useful for understanding the y-intercept of the parabola (which is at \( c \)), helping in initial plotting on the graph.
You can convert from vertex form to standard form by expanding and simplifying the vertex-form equation. This is useful for more detailed algebraic manipulation and when solving quadratic equations.
Determining 'a' in Parabola Equations
The value \( a \) in both the vertex and standard forms is crucial for defining the shape and orientation of a parabola.
To determine \( a \), you need:
  • The vertex of the parabola
  • Another point that the parabola passes through
Using the exercise as an example, start with the vertex form equation:\( y = a(x - h)^2 + k \). By substituting the vertex values \((h, k)\) into the formula, you simplify it. Then, by putting in the coordinates of another known point (x, y), you can solve for \( a \).
For instance, substituting the point (2, 3) into \( y = a(x - 4)^2 - 1 \), you derive \( 3 = 4a - 1 \). Solving this equation gives \( a = 1 \).
Knowing \( a \) helps finalize the equation of a parabola, giving a complete understanding of its graphing behavior.

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Most popular questions from this chapter

Find the rational zeros of the polynomial function. $$f(z)=z^{3}+\frac{11}{6} z^{2}-\frac{1}{2} z-\frac{1}{3}=\frac{1}{6}\left(6 z^{3}+11 z^{2}-3 z-2\right)$$

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