Quadratic functions and linear functions are quite different when you look at how they are graphed. Let's break it down:
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Shape:
- Linear Functions: These are shown with the equation (y = mx + c). Here, (m) is the slope (how steep the line is) and (c) is where the line crosses the y-axis. The graph looks like a straight line.
- Quadratic Functions: These use the equation (y = ax^2 + bx + c). Their graph looks like a U-shaped curve, called a parabola. It can open up (when (a > 0)) or down (when (a < 0)).
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Key Features:
- Linear Functions: They have endless solutions and cross the y-axis at just one point.
- Quadratic Functions: They have a special point called the vertex, which is the highest or lowest point on the graph. They can touch the x-axis at 0, 1, or 2 points. Whether they do that depends on something called the discriminant, which is found using (b^2 - 4ac).
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Graph Behavior:
- Linear: The change is steady. This means if you move along the line, it changes at the same rate.
- Quadratic: The change is not steady. It varies and the graph is symmetric, which means it looks the same on both sides of the vertex.
Understanding these differences can help you see how each type of function behaves and appears on a graph!