It should look like a parabola with the part below x-axis cut off. the left side starts from x=-1, opening upward & to the left, the right side starts from x=2, opening upward and to the right.
Answer: the function g(x) has the smallest minimum y-value.
Explanation:
1) The function f(x) = 3x² + 12x + 16 is a parabola.
The vertex of the parabola is the minimum or maximum on the parabola.
If the parabola open down then the vertex is a maximum, and if the parabola open upward the vertex is a minimum.
The sign of the coefficient of the quadratic term tells whether the parabola opens upward or downward.
When such coefficient is positive, the parabola opens upward (so it has a minimum); when the coefficient is negative the parabola opens downward (so it has a maximum).
Here the coefficient is positive (3), which tells that the vertex of the parabola is a miimum.
Then, finding the minimum value of the function is done by finding the vertex.
I will change the form of the function to the vertex form by completing squares:
Given: 3x² + 12x + 16
Group: (3x² + 12x) + 16
Common factor: 3 [x² + 4x ] + 16
Complete squares: 3[ ( x² + 4x + 4) - 4] + 16
Factor the trinomial: 3 [(x + 2)² - 4] + 16
Distributive property: 3 (x + 2)² - 12 + 16
Combine like terms: 3 (x + 2)² + 4
That is the vertex form: A(x - h)² + k, whch means that the vertex is (h,k) = (-2, 4).
Then the minimum value is 4 (when x = - 2).
2) The othe function is <span>g(x)= 2 *sin(x-pi)
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The sine function goes from -1 to + 1, so the minimum value of sin(x - pi) is - 1.
When you multiply by 2, you just increased the amplitude of the function and obtain the new minimum value is 2 (-1) = - 2
Comparing the two minima, you have 4 vs - 2, and so the function g(x) has the smallest minimum y-value.
Answer:
The number of Pumpkin can used = 48 cans
The number of Green Beans can used = 16 cans
Step-by-step explanation:
The ratio of Pumpkin to Green Beans = 3 : 1
So, for every 3 can of Pumpkin, 1 can of green beans is used.
Now, the total number of John wants to use = 64
Let us assume the common factor in the ratio is m.
⇒ The number of Pumpkin can used = 3 m
The number of Green Beans can used = 1 m = m
So, the Total cans used = Number of (Pumpkin + Green Beans) can
⇒ 3 m + m = 64
or, 4 m = 64
⇒ m = 64/ 4 = 16
Hence, The number of Pumpkin can used = 3 m = 3 x 16 = 48 cans
The number of Green Beans can used = m = 16 cans
