Explain how you could write a quadratic function in factored form that would have a vertex with an x-coordinate of 3 and two dis
tinct roots.
2 answers:
Response: The solution is f(x) = (x-3)²-h = (x-3-√h)(x-3+√h).
Detailed explanation: We are tasked with writing a quadratic equation in factored format that has a vertex at an x-coordinate of 3 and two separate roots.
A quadratic function with a vertex at x-coordinate k is represented in the form of a parabola:
In this case, both 'k' and 'h' are real numbers.
Given that the vertex's x-coordinate is 3, we have k = 3.
Consequently, the quadratic function can be expressed as
This represents the required factored form for the quadratic equation.
Refer to the included graph where the vertex has an x-coordinate of 3, and h is assumed to be 2 units.
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Answer:
(a) P(12.99 ≤ X ≤ 13.01) = 0.3840
(b) P(X ≥ 13.01) = 0.3075
Step-by-step explanation:
To address this problem, one needs to grasp the normal probability distribution and the central limit theorem.
Normal probability distribution
Problems involving normally distributed samples are resolved through the application of the z-score formula.
In a data set with a mean and standard deviation
, the z-score for a measurement X is defined as:
The Z-score indicates how many standard deviations the measurement deviates from the mean. After determining the Z-score, one can consult the z-score table to find the relevant p-value for that score. This p-value represents the probability that the measure's value is less than X, which indicates the percentile of X. By subtracting this p-value from 1, we obtain the likelihood that the measure's value exceeds X.
Central Limit Theorem
According to the Central Limit Theorem, for a normally distributed random variable X with mean and standard deviation
, the sampling distribution of the sample means with size n can be approximated by a normal distribution, characterized by mean
and standard deviation
.
The Central Limit Theorem can also be applicable to a skewed variable, provided that n is 30 or more.
In this case, we are given that:
(a) Compute P(12.99 ≤ X ≤ 13.01) when n = 16.
In this scenario, we find
This probability equates to the p-value of Z at X = 13.01 minus the p-value of Z at X = 12.99.
X = 13.01
According to the Central Limit Theorem
yields a p-value of 0.3075
0.6915 - 0.3075 = 0.3840
P(12.99 ≤ X ≤ 13.01) = 0.3840
(b) What is the probability that the sample mean diameter is greater than 13.01 when n = 25?
P(X ≥ 13.01) =
This is obtained by subtracting the p-value of Z when X = 13.01 from 1. Hence
has a p-value of 0.6915
1 - 0.6915 = 0.3075
P(X ≥ 13.01) = 0.3075
Answer: To remove fractions prior to solving, each term in the equation must be multiplied by .
Step-by-step explanation:
Consider the given expression:
It is essential to simplify this before attempting to solve it.
Since the denominators differ, identifying the Least Common Denominator (LCD) is necessary.
Break down the denominators into their prime components:
Select , as it possesses the greatest exponent. Thus:
Ultimately, to remove the fractions before solving, multiply both sides by 4: