Sketch the graph of a quadratic function for which the related quadratic equation has only one solution. Then, sketch one that has two solutions. Explain how you know by looking at the graph the number of solutions there are.
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To sketch the graph of a quadratic function, we start by considering the general form of a quadratic equation: y = ax^2 + bx + c. The number of solutions to the quadratic equation can be determined by looking at the discriminant (b^2 - 4ac).
For a quadratic equation with only one solution, the discriminant must equal zero. This occurs when the equation is in the form ax^2 + bx + c = 0, and the graph of the corresponding quadratic function will intersect the x-axis at only one point.
To illustrate this, let's consider the function y = x^2. This quadratic equation has only one solution when x = 0. By plotting several points on the graph, we can observe that the parabola intersects the x-axis at only one point:
(0, 0) (1, 1) (-1, 1)
The graph of y = x^2 is a upward-opening parabola that passes through the origin (0,0) and has symmetry about the y-axis. As we can see, there is only one x-axis intersection (x = 0), indicating one solution to the quadratic equation.
Now, let's consider a quadratic function with two solutions. For this case, the discriminant of the quadratic equation must be greater than zero. This occurs when the equation is in the form ax^2 + bx + c = 0, and the graph of the corresponding quadratic function will intersect the x-axis at two distinct points.
For example, let's look at the function y = x^2 - 4. By plotting points on the graph, we can observe that the parabola intersects the x-axis at two distinct points:
(-2, 0) (2, 0) (0, -4)
The graph of y = x^2 - 4 is an upward-opening parabola that intersects the x-axis at x = -2 and x = 2. This indicates two solutions to the quadratic equation.
In summary, by analyzing the graph of a quadratic function, we can determine the number of solutions to the related quadratic equation. If the graph intersects the x-axis at one point, there is one solution. If the graph intersects the x-axis at two distinct points, there are two solutions.