Find the coordinates of the points of intersection of the straight line y = 3x - 6 with the parabola y = x^2 - 6x + 8 and solve Graphically.
to solve algebraically, just solve
3x-6 = x^2-6x+8
x^2-9x+14 = 0
(x-7)(x+2) = 0
The graphical solution is at
http://www.wolframalpha.com/input/?i=solve+x^2-6x%2B8+%3D+3x-6
To find the coordinates of the points of intersection between the line and the parabola graphically, you can follow these steps:
Step 1: Graph the Line
To graph the line y = 3x - 6, you can plot a few points and draw a straight line passing through them. Choose a range of x-values and calculate the corresponding y-values using the equation of the line.
For example, if we choose x = -2, -1, 0, 1, and 2, we can calculate the corresponding y-values using the equation y = 3x - 6:
When x = -2, y = 3(-2) - 6 = -12
When x = -1, y = 3(-1) - 6 = -9
When x = 0, y = 3(0) - 6 = -6
When x = 1, y = 3(1) - 6 = -3
When x = 2, y = 3(2) - 6 = 0
Plot these points on a graph and draw a straight line connecting them.
Step 2: Graph the Parabola
To graph the parabola y = x^2 - 6x + 8, you can use the vertex form of a parabola, which is given by y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
The vertex of this parabola can be found by using the formula h = -b/2a, where a = 1, b = -6, and c = 8.
In this case, h = -(-6)/(2*1) = 3.
To find the y-coordinate of the vertex, substitute h back into the equation of the parabola:
k = h^2 - 6h + 8 = 3^2 - 6(3) + 8 = 9 - 18 + 8 = -1.
So, the vertex of the parabola is (3, -1).
Based on the vertex and the shape of the parabola (which is open upward since a = 1 > 0), you can plot some additional points on the graph to get a general idea of the shape and direction of the parabola.
Step 3: Find the Points of Intersection
Now, you can observe the graph and identify the points where the line and the parabola intersect. These points will have the same coordinates on both graphs.
Look for the points where the line and the parabola intersect on the graph. You can estimate the approximate coordinates of these points by reading them off the graph.
For instance, one point of intersection might be around (1, -3). Repeat this process to find another point of intersection.
Step 4: Verify the Coordinates
To verify the coordinates of the points of intersection, substitute the x and y values into the equations of the line and the parabola and check if they satisfy both equations.
For example, if you found a point of intersection at (1, -3), substitute x = 1 and y = -3 into both equations:
For the line equation: -3 = 3(1) - 6 = 3 - 6 = -3 (satisfied).
For the parabola equation: -3 = 1^2 - 6(1) + 8 = 1 - 6 + 8 = 3 (not satisfied).
In this case, (1, -3) is not a valid point of intersection.
Repeat steps 3 and 4 with different points of intersection obtained from the graph until you find the correct coordinates of the points of intersection.
Note: Although the graphical method can give approximate solutions, it may not always be precise. For more accurate results, you can use algebraic methods such as substitution or elimination to solve the system of equations formed by equating the line and the parabola equations.