If I were looking at a graph of a quadratic equation, how do I determine where the solutions are?
if it is a function like
ax^2 + bx + c = y
then the solutions are when y=0, i.e. where the line crosses the x axis.
Thank you! Is this case for all functions?
If b^2<4ac the solutions are imaginary.
if b^2=4ac there is only one solution
where
x=-b/2a
Okay I think I understand now. Thank you!
To determine the solutions of a quadratic equation from its graph, you need to find the x-values where the graph intersects or touches the x-axis. These points on the x-axis correspond to the roots or solutions of the equation.
To find these solutions visually from a graph, follow these steps:
1. Identify the vertex: The vertex is the highest or lowest point on the graph. It lies on the axis of symmetry of the parabola.
2. Draw a horizontal line from the vertex to the y-axis: The point where this line intersects the y-axis is the y-coordinate of the vertex, which corresponds to the value of the quadratic equation at the x-coordinate of the vertex.
3. Locate the x-intercepts: The x-intercepts are the points where the graph intersects or touches the x-axis. These points represent the solutions of the quadratic equation.
If the parabola intersects the x-axis at two distinct points, then the quadratic equation has two real solutions. If the parabola just touches the x-axis at one point, then the quadratic equation has one real solution (a double root). And if the parabola does not intersect or touch the x-axis, then the quadratic equation has no real solutions (complex roots).
Note: The shape and position of the parabola (whether it opens upwards or downwards) can also provide additional information about the solutions, but the x-intercepts are the primary points to consider.