use a graphing calculator to graph the function y=x^2-3x+15. a. what does the graph let you conclude about real number solutions of x^2-4x+13=0? explain. b. substitue 2-2i for x in the equation x^2-4x+13=0. simplify. is the resulting equation true? show your work.
To graph the function y=x^2-3x+15 on a graphing calculator, follow these steps:
1. Turn on your graphing calculator.
2. Enter the equation y=x^2-3x+15 into the graphing calculator.
3. Press the corresponding function key to graph the equation.
a. The graph of the function y=x^2-3x+15 provides information about the real number solutions of the equation x^2-4x+13=0. By observing the graph, we can conclude the following:
- The graph of y=x^2-3x+15 is a parabola.
- The vertex of the parabola can be determined by finding the x-coordinate of the vertex, which is given by the formula x=-b/(2a). In this case, a=1 and b=-3, so the x-coordinate of the vertex is x=-(-3)/(2*1)=3/2.
- Since the coefficient of x^2 term (a=1) is positive, the parabola opens upward.
- As a result, the parabola does not intersect the x-axis and does not have real number solutions for the equation x^2-4x+13=0. In other words, there are no real number solutions for the equation x^2-4x+13=0.
b. To substitute 2-2i for x in the equation x^2-4x+13=0, we replace every occurrence of x with 2-2i:
(2-2i)^2 - 4(2-2i) + 13 = 0
Simplifying:
(2-2i)(2-2i) - 4(2-2i) + 13 = 0
(4-4i-4i+4i^2) - (8-8i) + 13 = 0
(4-8i+4i^2) - 8 + 8i + 13 = 0
(4-8i-4) + 8i + 13 = 0
0 + 8i + 13 = 0
8i + 13 = 0
8i = -13
So, the resulting equation is 8i = -13. This equation is not true since 8i is not equal to -13.
To graph the function, y = x^2 - 3x + 15, follow these steps using a graphing calculator:
1. Turn on your graphing calculator and go to the graphing mode.
2. Enter the function by pressing the appropriate buttons. In this case, enter y = x^2 - 3x + 15.
3. Set the window settings to appropriately view the graph. Use the range for x and y values that you prefer. For example, you may set the x-axis from -10 to 10 and the y-axis from -10 to 100.
4. Graph the function by selecting the appropriate graphing command (usually, the "graph" button).
Once you have the graph displayed, you can analyze it to answer the given questions:
a. To find conclusions about the real number solutions of x^2 - 4x + 13 = 0 from the graph, look for the x-intercepts (where the graph intersects the x-axis). If the graph intersects the x-axis at any point(s), then there exist real number solutions to the equation. If the graph does not intersect the x-axis at any point, then there are no real number solutions.
b. To substitute 2 - 2i for x in the equation x^2 - 4x + 13 = 0, replace x with 2 - 2i and simplify the equation:
(2 - 2i)^2 - 4(2 - 2i) + 13 = 0
Expanding and simplifying the equation, we get:
(2 - 2i)(2 - 2i) - 8 + 8i + 13 = 0
Simplifying further:
4 - 4i - 4i + 4i^2 - 8 + 8i + 13 = 0
Combining like terms and using the definition of i^2 = -1:
-11 - 4i + 4(-1) + 8i = 0
-11 - 4i - 4 + 8i = 0
-15 + 4i = 0
This resulting equation, -15 + 4i = 0, is not true since it does not equal zero.