graph the following function using transformations, be sure to graph all of the stages one graph, state the domain and the range y=x^2+12x
To graph the function y = x^2 + 12x using transformations, let's break it down into its individual stages: shifting, scaling, and reflection.
1. Shifting: The first step is to identify the vertex of the parabola, which represents its lowest or highest point. For a quadratic function in the form of y = ax^2 + bx + c, the vertex can be found using the formula: (-b/2a, f(-b/2a)). In our case, a = 1, b = 12, and c = 0, so the x-coordinate of the vertex can be found at -12/2*1 = -6.
2. Scaling: The coefficient "a" determines the scaling factor for the parabola. A positive value for "a" indicates the parabola opens upwards, while a negative value indicates it opens downwards. Since a = 1 in our case, the parabola opens upwards.
3. Reflection: The parabola remains unreflected with respect to the y-axis. Therefore, there is no reflection.
Now, let's proceed to graphing the function:
1. Domain: Since the function is a quadratic equation, the domain is all real numbers (-∞, +∞).
2. Range: Since the coefficient of the x^2 term is positive, the parabola opens upwards. Thus, the minimum value is the y-coordinate of the vertex, and the range is [y-coordinate of the vertex, +∞). In our case, y = -6^2 + 12*(-6) = -36 - 72 = -108. Therefore, the range is (-108, +∞).
To graph the function, follow these steps:
Step 1: Plot the vertex. In our case, the vertex is at (-6, -108).
Step 2: Determine other key points. Choose some x-values to find their corresponding y-values using the equation y = x^2 + 12x. For example, when x = -9, y = (-9)^2 + 12*(-9) = 81 - 108 = -27. Similarly, when x = -3, y = (-3)^2 + 12*(-3) = 9 - 36 = -27. Another point to consider is the x-intercept, which can be found by setting y = 0 and solving for x.
Step 3: Draw the parabola. Connect the points obtained in the previous step, including the vertex, the x-intercepts (if any), and any other selected points.
The resulting graph should show a parabola that opens upwards, with its vertex at (-6, -108).