Problem 54
Question
Describe how to find a parabola's vertex if its equation is expressed in standard form. Give an example.
Step-by-Step Solution
Verified Answer
For a parabola expressed in standard form \(y = a(x - h)^2 + k\), the vertex is given by the coordinates (h, k). In the example equation \(y = 2(x - 3)^2 + 4\), the vertex is (3, 4).
1Step 1: Understanding the standard form
A parabola in its standard form is expressed as \(y = a(x - h)^2 + k\). In this equation, 'a' determines the shape of the parabola, either opening upwards or downwards. The vertex of the parabola is represented by the coordinates (h, k).
2Step 2: Identifying 'h' and 'k'
In the standard equation, (h, k) are the values that constitute the vertex. Therefore, to figure out the parabola's vertex, simply identify the values of 'h' and 'k' in the equation.
3Step 3: Example
Take for example the equation \(y = 2(x - 3)^2 + 4\). To determine the vertex of this parabola, identify 'h' and 'k'. Here, 'h' equals to 3 and 'k' equals to 4. Therefore, the vertex is (3, 4).
Other exercises in this chapter
Problem 54
Suppose \(\frac{3}{4}\) is a root of a polynomial equation. What does this tell us about the leading coefficient and the constant term in the equation?
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If you know that \(-2\) is a zero of $$f(x)=x^{3}+7 x^{2}+4 x-12$$ explain how to solve the equation $$x^{3}+7 x^{2}+4 x-12=0$$
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Describe in words the variation shown by the given equation. \(z=\frac{k \sqrt{x}}{y^{2}}\)
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In Exercises \(55-56,\) use a graphing utility to determine upper and lower bounds for the zeros of \(f .\) Does synthetic division verify your observations? $$
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