Quadratic functions
How do three coefficients transform a curve?
One comparison at a time
Experiment with a guide
Connect the height of the curve to the points where it touches the horizontal axis.
Comparison 1 of 3
Find two roots
We start with f(x) = x² − 4x + 3. A root is an input that gives zero height. Follow the curve to its two crossings of the x axis.
The points at x = 1 and x = 3 have zero height. The vertex lies below the axis.
You are exploring other values. You can continue freely or apply this comparison’s values again.
How to read this drawing
The x axis is horizontal: choose an input there. Height on the y axis is f(x), the result. Points mark roots (zero height) and the vertex (turning point). The graph reframes itself: compare the axis numbers, not only the size on screen.
From drawing to graph
What changes? Let’s compare.
A line shows the relationship between quantities. The point marks your choice or the instant in the animation. First read the axis names and units.
The graph and table on the right show the same x chosen above.
One experiment at a time
Save. Change. Compare.
Save the current values, change one quantity and observe both lines on the same graph. The saved line is dashed. The attempt stays in this browser.
Compare the chosen values
What if… Start with a = 1, b = −4 and c = 3. Find the roots, then change only c.
What is happening?
Each coefficient has a role: a controls the direction of opening, b influences the position of the axis of symmetry, and c gives the vertical intercept.
If a is positive, the parabola opens upwards; if it is negative, it opens downwards. The vertex is where the function reaches its minimum or maximum. Real roots are the x-values where the curve crosses or touches the horizontal axis. When a = 0, the function is no longer quadratic: it becomes linear or constant.
a ≠ 0 — quadratic function · Δ = b² − 4ac
The model and its limits
The graph shows a finite interval of x and adjusts its view to include the vertex and roots. When a = b = c = 0, every real number is a root.