Question 638522: 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+ 1 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
done,
see your previous post of the same question
To graph the function y = x^2 + 12x using transformations, we need to understand the effects of each individual transformation: shifting, stretching, and reflecting. Here are the steps to graph this function:
Step 1: Start with the parent function y = x^2
This is the basic quadratic function, which will serve as our starting point.
Step 2: Observe the coefficient of x^2, which is 1.
Since it is positive, the parabola will open upwards. If it were negative, the parabola would open downwards.
Step 3: Identify the transformation due to the x-term, which is 12x.
The coefficient of x, which is 12, tells us that the parabola will shift horizontally.
Step 4: Calculate the axis of symmetry.
The axis of symmetry can be found by using the formula: x = -b/2a.
In this case, a = 1 and b = 12.
So, x = -12/2(1) = -12/2 = -6.
Step 5: Identify the vertex.
The x-coordinate of the vertex is the same as the axis of symmetry, which is -6.
To find the y-coordinate, substitute -6 into the original function: y = (-6)^2 + 12(-6).
Simplifying this equation gives us y = 36 - 72 = -36.
Therefore, the vertex is (-6, -36).
Step 6: Determine the y-intercept.
The y-intercept is the value of y when x = 0.
Substituting x = 0 into the original equation, we get y = (0)^2 + 12(0) = 0.
Therefore, the y-intercept is 0.
Step 7: Plot the vertex, y-intercept, and the points symmetric to the vertex.
Using the calculated values, plot these points on a graph.
Step 8: Draw the parabola.
Based on the information obtained, we can sketch the parabolic shape by connecting the plotted points.
Step 9: Determine the domain and range.
The domain is the set of all possible x-values for which the function is defined. In this case, the domain is all real numbers since there are no restrictions on the x-values.
The range is the set of all possible y-values. Since the parabola is opening upwards, the minimum value occurs at the vertex (-6, -36). Therefore, the range is all real numbers greater than or equal to -36.
By following these steps, you should be able to accurately graph the given function y = x^2 + 12x and determine the domain and range.