Find the minimum or maximum value. Explain how you know whether it is a minimum or maximum.
𝑦 =2/3 𝑥^2 − 6𝑥 + 4
To find the minimum or maximum value of the given function y = (2/3)x^2 - 6x + 4, we can first observe that the coefficient of the x^2 term is positive (2/3), indicating that the parabola opens upwards.
Since the parabola opens upwards, the vertex of the parabola represents the minimum value of the function. We can find the x-coordinate of the vertex using the formula x = -b/2a, where a and b are the coefficients of the x^2 and x terms respectively.
In this case, a = 2/3 and b = -6. Plugging these values into the formula, we have x = -(-6) / (2/3), which simplifies to x = 9.
To find the y-coordinate of the vertex, we substitute the x-coordinate into the function y = (2/3)x^2 - 6x + 4:
y = (2/3)(9)^2 - 6(9) + 4
y = 6(27) - 54 + 4
y = 162 - 54 + 4
y = 112
Therefore, the minimum value of y is 112, and it occurs when x = 9.
To find the minimum or maximum value of the given function 𝑦 = 2/3 𝑥^2 − 6𝑥 + 4, we can use a concept in calculus called the vertex form of a quadratic equation. The vertex form of a quadratic equation is 𝑦 = 𝑎(𝑥 − ℎ)^2 + 𝑘, where (ℎ, 𝑘) represents the coordinates of the vertex.
Step 1: Identify the coefficients 𝑎, 𝑏, and 𝑐 from the given equation 𝑦 = 2/3 𝑥^2 − 6𝑥 + 4.
In this case, 𝑎 = 2/3, 𝑏 = -6, and 𝑐 = 4.
Step 2: Calculate the x-coordinate of the vertex using the formula ℎ = -𝑏 / 2𝑎.
Plugging in the values, we get ℎ = -(-6) / (2 * 2/3) = 9.
Step 3: Substitute the x-coordinate of the vertex back into the original equation to find the y-coordinate of the vertex.
Plugging in ℎ = 9 into the equation 𝑦 = 2/3 𝑥^2 − 6𝑥 + 4, we get 𝑦 = 2/3(9)^2 − 6(9) + 4 = -1.
Therefore, the vertex of the function 𝑦 = 2/3 𝑥^2 − 6𝑥 + 4 is (9, -1).
To determine whether it is a minimum or maximum value, we can look at the coefficient 𝑎. If 𝑎 > 0, the parabola opens upward, indicating a minimum value at the vertex. If 𝑎 < 0, the parabola opens downward, indicating a maximum value at the vertex.
In this case, 𝑎 = 2/3 > 0, so the parabola opens upward, and the vertex at (9, -1) represents the minimum value of the function 𝑦 = 2/3 𝑥^2 − 6𝑥 + 4.