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.
A quadratic with one solution has intercepts on the x axis that are the same. y= x^2-1 is an example. If it has two solutions, it will be of the form y= x^2+2x -.2, and have two intercepts.
Plot these and make certain you understand it.
To sketch the graph of a quadratic function, we need to consider the general form of a quadratic function: f(x) = ax^2 + bx + c, where a, b, and c are constants.
For a quadratic equation to have only one solution, it means that the graph of the corresponding quadratic function will have a single x-intercept or a single point where it intersects the x-axis. This can happen when the graph is a perfect square or a perfect zoom-in of a parabola that does not cross the x-axis. In other words, if the vertex of the parabola is right on the x-axis.
To create such a graph, we can start by setting the discriminant (b^2 - 4ac) of the quadratic equation to zero since it will yield a single solution. If the discriminant is zero, it means that the equation will have only one x-intercept.
Now, let's consider two scenarios:
1. One Solution (Discriminant = 0):
Suppose we have the quadratic equation ax^2 + bx + c = 0 with a discriminant of (b^2 - 4ac) = 0. To create a graph with only one solution, we can choose values for a, b, and c such that the discriminant is zero. For example, let's say a = 1, b = -2, and c = 1. Plugging these values into the equation, we get x^2 - 2x + 1 = 0.
To find the solution, we can factor this equation into (x - 1)^2 = 0, which means that x = 1. This implies that the graph will touch the x-axis at x = 1 and will not cross or intersect it in any other location. The vertex of this parabola will be at (1, 0).
Graphically, you will see a parabola opening upwards, reaching the x-axis at x = 1, and curving back up. The vertex will be at the point (1, 0).
2. Two Solutions (Discriminant > 0):
Suppose now we have a quadratic equation ax^2 + bx + c = 0 with a discriminant (b^2 - 4ac) > 0. To create a graph with two solutions, we need to choose values for a, b, and c such that the discriminant is positive. For example, let's say a = 1, b = -3, and c = 2. Plugging these values into the equation, we get x^2 - 3x + 2 = 0.
To find the solutions, we can factor this equation into (x - 1)(x - 2) = 0, which means that x = 1 or x = 2. This implies that the graph will intersect the x-axis at both x = 1 and x = 2. The vertex of this parabola will be in between the two x-intercepts.
Graphically, you will see a parabola opening upwards, intersecting the x-axis at x = 1 and x = 2, and curving back up in between. The vertex will be located halfway between the two x-values.
In summary, by looking at the graph of a quadratic function, we can determine the number of solutions (x-intercepts or points of intersection with the x-axis). If the graph has a single x-intercept, there will be one solution to the corresponding quadratic equation. If the graph has two x-intercepts, there will be two solutions to the quadratic equation.