How do you graph a quadratic? Example: y = x² − 4x + 3
Find the vertex and the intercepts, then draw a U shape through them. For y = x² − 4x + 3 the calculator labels these points.
Since a = 1 is positive, the parabola opens upward and the vertex is its lowest point. The mirror image of (0, 3) across x = 2 is (4, 3), which gives you one more point for free.
How do you find the intercepts of a polynomial graph?
The y-intercept is the constant term, because every other term is zero at x = 0. The x-intercepts are the real solutions of f(x) = 0.
For y = x³ − 4x the y-intercept is (0, 0). Factoring gives x(x − 2)(x + 2), so the curve crosses the x-axis at −2, 0, and 2. For higher degrees the graph labels the y-intercept, and the zeros calculator gives the exact x-intercepts.
What does end behavior mean on a polynomial graph?
End behavior is what the graph does at the far left and far right. Only the degree and the sign of the leading coefficient decide it.
| Degree | Leading coefficient | Ends of the graph |
|---|---|---|
| Even (2, 4, 6) | Positive | Both ends rise |
| Even (2, 4, 6) | Negative | Both ends fall |
| Odd (3, 5) | Positive | Falls on the left, rises on the right |
| Odd (3, 5) | Negative | Rises on the left, falls on the right |
So y = x⁴ − 5x² + 4 rises on both sides, while y = −x³ + 2x starts high on the left and ends low on the right. The middle of the graph can wiggle, but a polynomial of degree n turns at most n − 1 times.
How do you change the graph window?
Edit the X min, X max, Y min, and Y max boxes under the graph and press Redraw. The default view runs from −10 to 10 on both axes.
If a vertex or turning point is off the screen, widen the y-range. If the interesting part is squeezed into the middle, narrow the x-range to zoom in. Changing the window changes only the view; the function and its labeled points stay the same.
For y = x⁴ − 5x² + 4, the default window works well, since the curve dips to just below −2 near the middle. For something like y = x³ − 30x, the turning points are at about y = ±63, so set Y min to −80 and Y max to 80 to see them.
What can you enter in this graphing calculator?
Any polynomial in x up to degree 6, written as y = … or on its own. Use ^ for powers: y = x^3 - 4x or y = 2x^2 + x - 1.
It is a graphing calculator for polynomial functions, so it does not plot trig, logarithm, or rational functions. For two straight lines at once, the system of equations calculator graphs both lines and their intersection.
What are common mistakes when graphing polynomials?
Drawing a parabola with a pointed bottom is one. Polynomial graphs are always smooth, so the vertex is a rounded turn.
Getting end behavior backwards is another, especially with a negative leading coefficient. Students also plot a few points too close together, miss a turning point, and conclude the curve is a straight line. Checking the degree first tells you how many turns to look for.
Why does a graph bounce off the x-axis at some roots?
A root that appears an even number of times makes the graph touch the axis and turn back. A root that appears an odd number of times makes it cross.
y = (x − 1)² touches the x-axis at x = 1 and stays above it, because the factor appears twice. y = x³ − 4x crosses at all three of its roots, since each factor appears once. Looking at the factored form tells you which kind of root you have before you draw anything.
How do you check an answer with a graph?
Graph the polynomial and look at where it crosses the x-axis. Those x-values should match the solutions you found for f(x) = 0.
Say you solved x² − 4x + 3 = 0 and got x = 1 and x = 3. Graph y = x² − 4x + 3 and you will see the curve cross at exactly those two points. If your answers were 2 and 4 instead, the graph would show the mistake at once. This works for any degree up to 6, which makes the graph a fast second opinion on homework.