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Magazine Chapter 13: Problem 42
Write each equation of a parabola in standard form and graph it. Give the coordinates of the vertex. $$ y=-2 x^{2}-4 x $$
Short Answer
Expert verified
The standard form is \( y = -2(x+1)^2 + 2 \) with the vertex at \((-1, 2)\).
Step by step solution
01
Identify the Original Equation
The given equation of the parabola is given as \( y = -2x^2 - 4x \). It is currently expressed in general quadratic form \( Ax^2 + Bx + C \), where \( A = -2 \), \( B = -4 \), and \( C = 0 \).
02
Rewrite in Standard Form
The standard form of a parabola's equation is \( y = a(x-h)^2 + k \). To convert the given equation into this form, complete the square. Start by factoring the coefficient of the \( x^2 \) term from the \( x \) terms: \( y = -2(x^2 + 2x) \).
03
Complete the Square
Within the parenthesis \((x^2 + 2x)\), take half of the \( x \)-coefficient (which is 2) and square it: \( \left( \frac{2}{2} \right)^2 = 1 \). Add and subtract this inside the parenthesis: \( y = -2(x^2 + 2x + 1 - 1) \) which simplifies to \( y = -2((x + 1)^2 - 1) \).
04
Simplify the Equation
Distribute the \( -2 \) across the binomial: \( y = -2(x + 1)^2 + 2 \). This is the standard form of the parabola, with \( a = -2 \), \( h = -1 \), and \( k = 2 \).
05
Identify the Vertex
From the standard form \( y = -2(x+1)^2 + 2 \), the vertex \((h, k)\) is at \((-1, 2)\).
06
Graph the Parabola
Plot the vertex \((-1, 2)\) on the coordinate plane. Since the \( a \) value is negative, the parabola opens downward. Sketch the parabola to visually represent these characteristics with the axis of symmetry at \( x = -1 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Standard Form of a Parabola
The **standard form of a parabola** is a way to express the quadratic equation of a parabola in a neat and useful format. When you have a quadratic equation like \( y = ax^2 + bx + c \), converting it to standard form makes it easier to understand and graph the parabola. The standard form is given as:
- \( y = a(x-h)^2 + k \)
- \( a \) determines the width and the direction of the opening: if \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.
- \( h \) and \( k \) are the coordinates of the vertex, which is the highest or lowest point of the parabola depending on its orientation.
Completing the Square
**Completing the square** is a valuable algebraic method used to rewrite a quadratic expression in a more useful form. This method is crucial when you need to convert a quadratic equation into the standard form. It involves adding and subtracting the same value within a quadratic expression to make it a perfect square trinomial.Here's how you do it:
- Take the quadratic equation and group the terms involving \( x \) together.
- Factor out any coefficient in front of \( x^2 \), if necessary.
- Take the coefficient of the \( x \) term, divide it by 2, and then square it.
- Add and subtract this square inside the group of \( x \) terms.
- Finally, write the terms as a perfect square trinomial and simplify the equation.
Vertex of a Parabola
The **vertex of a parabola** is a significant feature. It's the point that represents the maximum or minimum value of the quadratic function. In the standard form, \( y = a(x-h)^2 + k \), the vertex of the parabola can be directly read as \((h, k)\).Knowing how to find the vertex is crucial because:
- It gives us the point where the parabola turns direction, which is the maximum if the parabola opens downward (\( a < 0 \)), or the minimum if it opens upward (\( a > 0 \)).
- This point also forms the axis of symmetry of the parabola, meaning the graph is a mirror image on either side of the line \( x = h \).
