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Chapter 3: Problem 77

Describe how to find a parabola's vertex if its equation is expressed in standard form. Give an example.

Short Answer

Expert verified
The vertex of a parabola represented in standard form \(y = ax^2 + bx + c\) is given by \((h, k)\) where \(h = -b/2a\) and \(k\) is the result of substituting \(h\) back into the original equation. For example, for the equation \(y = 2x^2 - 4x + 2\), the vertex is (1, 0).

Step by step solution

01

Understanding Standard Form

A standard form of a quadratic equation is given as \(y = ax^2 + bx + c\). Here, 'a' is the coefficient of \(x^2\), 'b' is the coefficient of 'x', and 'c' is a constant.
02

Vertex Formula

The vertex formula \((h, k)\) for a parabola given by the equation \(y = ax^2 + bx + c\) is derived from completing the square. The x-coordinate of the vertex 'h' can be found using \(-b/2a\). Substituting 'h' in the original equation results in 'k', the y-coordinate of the vertex.
03

Application of the Vertex Formula

For example, let's take a quadratic equation \(y = 2x^2 - 4x + 2\). Here, 'a' is 2 and 'b' is -4. By applying the vertex formula, the 'h' value is given by \(-(-4)/(2*2)\) which is 1. Substitute 'h' (which is 1) into the quadratic equation to find 'k'. Thus, \(y = 2*1^2 - 4*1 + 2 = 0\). So, the vertex of the parabola is (1, 0).
04

Conclusion

Therefore, by using the vertex formula \(-b/2a\) and substituting it back into the quadratic equation, the vertex of the parabola can be found.

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Most popular questions from this chapter

The average number of daily phone calls, \(C\), between two cities varies jointly as the product of their populations, \(P_{1}\) and \(P_{2}\) and inversely as the square of the distance, \(d\), between them. a. Write an equation that expresses this relationship. b. The distance between San Francisco (population: \(777,000\) ) and Los Angeles (population: \(3,695,000\) ) is 420 miles. If the average number of daily phone calls between the cities is \(326,000,\) find the value of \(k\) to two decimal places and write the equation of variation. c. Memphis (population: \(650,000\) ) is 400 miles from New Orleans (population: \(490,000\) ). Find the average number of daily phone calls, to the nearest whole number, between these cities.

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