Write the following quadratic function in vertex form and state the vertex, the axis of symmetry, and the minimum or maximum value. 𝑦 = 3𝑥^2 + 24𝑥 + 41
To write the quadratic function in vertex form, we can complete the square.
First, let's factor out the common factor of 3 from the first two terms:
y = 3(x^2 + 8x) + 41
To complete the square, we need to add and subtract the square of half the coefficient of the x-term (in this case, 8/2 = 4):
y = 3(x^2 + 8x + 16 - 16) + 41
Now, we can group the first three terms and rewrite the expression:
y = 3((x^2 + 8x + 16) - 16) + 41
Next, we can simplify the expression inside the parentheses:
y = 3((x + 4)^2 - 16) + 41
To expand the expression, we distribute the 3 to both terms inside the parentheses:
y = 3(x + 4)^2 - 48 + 41
Now, we can combine like terms:
y = 3(x + 4)^2 - 7
Therefore, the quadratic function in vertex form is:
y = 3(x + 4)^2 - 7
The vertex form of the quadratic function is given by: y = a(x - h)^2 + k, where (h, k) is the vertex.
In this case, the vertex is (-4, -7).
The axis of symmetry is given by the equation: x = h, where h is the x-value of the vertex.
Therefore, the axis of symmetry is x = -4.
Since the coefficient of the x^2 term (a = 3) is positive, the parabola opens upwards and the vertex represents the minimum value.
Hence, the minimum value of the quadratic function is y = -7.