Graph the following function using transformations. Be sure to graph all of the stages on one
graph. State the domain and range.
For example, if you were asked to graph y= x^2+11 using transformations, you would show the
graph of y = x^2 and the graph shifted up 1 unit. Please do not show only the final graph.
Y=x^2+12x
you would complete the square ....
y = x^2 + 12x + 36 - 36
= (x+6)^2 - 36
so y= x^2 would be translated to the LEFT 6 units, an down 36
To graph the function y = x^2 + 12x using transformations, we will follow the steps below:
Step 1: Start by graphing the base function y = x^2.
- The base function y = x^2 is a simple quadratic curve that opens upward and passes through the point (0, 0).
Step 2: Apply the transformation of a vertical shift.
- The equation y = x^2 + 12x can be seen as the base function y = x^2 shifted vertically by adding a constant term of 12x.
- Since the coefficient of the x term is positive (12x), it means the graph will shift upward.
- To find the direction and amount of the vertical shift, we can calculate the vertex of the parabola. The x-coordinate of the vertex is given by -b/2a, where a and b are the coefficients in the quadratic equation (ax^2 + bx + c = 0).
- In this case, a = 1, and b = 12.
- Using the formula, we find the x-coordinate of the vertex: x = -12 / (2*1) = -6.
- Substitute this x-coordinate back into the equation to find the y-coordinate of the vertex: y = (-6)^2 + 12(-6) = 36 - 72 = -36.
- Therefore, the vertex is (-6, -36), indicating that we need to shift the graph of y = x^2 upward by 36 units.
Step 3: Graph the transformed function.
- Start with the graph of the base function y = x^2.
- Shift the graph vertically by moving all points on the graph 36 units upward. For example, the point (0, 0) on the base function will move to (0, -36) on the transformed graph.
- Continue shifting all other points on the base function by the same vertical amount of 36 units.
Domain and Range:
- The domain of the function y = x^2 + 12x is the set of all real numbers since any value of x can be squared and added to 12x.
- The range of the function will depend on the vertex of the parabola. Since the vertex is (-6, -36), the range will include all y-values less than or equal to -36.
Please note that the graph may not be accurate without specific scales and units on the axes. It's always useful to use graphing software or a graphing calculator to plot the graph precisely.