• If you are looking at a graph of a quadratic equation, how do you determine where the solutions are? Please provide and example.
To determine the solutions of a quadratic equation graphically, you need to examine the x-intercepts or roots, where the graph crosses the x-axis. The x-intercepts represent the values of x for which the equation equals zero.
Here's an example to illustrate the process:
Let's consider the quadratic equation: y = x^2 - 4x + 3
To find the solutions graphically, you can plot the equation on a coordinate plane and look for the points where the graph intersects the x-axis. The x-values of these points will be the solutions to the equation.
1. Plot the equation on a graph by assigning different values for x and calculating the corresponding y-values. For example, you could choose x-values ranging from -5 to 5.
When x = -5, y = (-5)^2 - 4(-5) + 3 = 33
When x = -4, y = (-4)^2 - 4(-4) + 3 = 27
...
When x = 0, y = (0)^2 - 4(0) + 3 = 3
...
When x = 5, y = (5)^2 - 4(5) + 3 = -2
2. Plot the points obtained from the calculations on the graph.
3. Connect the plotted points to sketch the graph of the quadratic equation.
4. Look for the points where the graph intersects or touches the x-axis. In this case, the graph intersects the x-axis at the points (1,0) and (3,0).
5. Therefore, the solutions to the quadratic equation y = x^2 - 4x + 3 are x = 1 and x = 3, since these are the x-values of the points where the graph intersects the x-axis.
By examining the graph and identifying the x-intercepts, you can determine the solutions of a quadratic equation visually.