Write the quadratic function 𝑦 = 6𝑥^2 − 24𝑥 − 12 in vertex form and state the characteristics of the function. Include the vertex, axis of symmetry, direction of opening, minimum or maximum value, domain, range, x- and y-intercepts, vertical stretch, horizontal translation, vertical translation, and mapping notation
To rewrite the quadratic function 𝑦 = 6𝑥^2 − 24𝑥 − 12 in vertex form, we need to complete the square. First, let's factor out the coefficient of 𝑥^2:
𝑦 = 6(𝑥^2 − 4𝑥) − 12
Now, let's complete the square by taking half of the coefficient of 𝑥, squaring it, and adding it inside the parentheses. In this case, half of -4 is -2, and when squared it is 4:
𝑦 = 6(𝑥^2 − 4𝑥 + 4 - 4) − 12
We have subtracted 4 inside the parentheses, so we need to add 4 back outside the parentheses to maintain the same equation:
𝑦 = 6(𝑥^2 − 4𝑥 + 4) - 12 - 24
Simplifying further:
𝑦 = 6(𝑥 - 2)^2 - 36
Now, the function is in vertex form, 𝑦 = 𝑎(𝑥 - ℎ)^2 + 𝑘, with the vertex at (ℎ, 𝑘). In this case, the vertex is (2, -36).
The axis of symmetry is given by the equation 𝑥 = ℎ, so the axis of symmetry is 𝑥 = 2.
Since the coefficient of 𝑥^2 is positive (6 > 0), the parabola opens upwards, so the direction of opening is upwards.
The vertex (2, -36) represents the minimum value of the function, which means the function has a minimum.
The domain of the function is all real numbers, as there are no restrictions on the value of 𝑥.
To determine the range, we consider that the parabola opens upwards, so the minimum value (-36) represents the lowest point on the graph. Thus, the range is all values greater than or equal to -36, or (-∞, -36].
To find the x-intercepts, we set 𝑦 = 0 and solve for 𝑥:
0 = 6(𝑥 - 2)^2 - 36
6(𝑥 - 2)^2 = 36
(𝑥 - 2)^2 = 6
Taking the square root of both sides:
𝑥 - 2 = ± √6
𝑥 = 2 ± √6
So the x-intercepts are (2 + √6, 0) and (2 - √6, 0).
To find the y-intercept, we set 𝑥 = 0 and solve for 𝑦:
𝑦 = 6(0)^2 - 24(0) - 12
𝑦 = -12
Thus, the y-intercept is (0, -12).
The vertical stretch is given by the coefficient 𝑎 of the quadratic term, which in this case is 6. Therefore, the vertical stretch is a factor of 6.
The horizontal translation is given by the constant in the vertex form equation, which in this case is (𝑥 - 2). The vertex form equation is 𝑦 = 𝑎(𝑥 - ℎ)^2 + 𝑘, where ℎ is the amount of horizontal translation. In this case, ℎ = 2, indicating that the graph is shifted right by 2 units.
There is no vertical translation because there is no constant term (ℎ) added to the vertex form equation.
Therefore, using mapping notation, we can express the function as:
𝑦 = 6(𝑥 - 2)^2 - 36
𝑦 = 𝑓(𝑥) = 6(𝑥 - 2)^2 - 36