A graph answers questions faster than algebra does, but only if you know which features to look for and how far to trust what you see. This guide walks through the five things a teacher usually asks about (roots, the y-intercept, turning points, asymptotes and intersections) using three functions you can type into the function grapher yourself. Every number below is what the grapher reports for the default view, x from −10 to 10, shown to six significant digits.
Start with the scale
Before reading anything, look at the axes. The grapher keeps one unit the same length on both axes, so a 45° line really is y = x and a circle looks round. Many textbook graphs and spreadsheet charts do not: they stretch y to fill the box, which makes a gentle slope look steep. Note the grid step too. If the grid lines are 2 units apart, a point halfway between two lines sits on an odd number, not on the next even one.
Roots: where the curve meets y = 0
A root (or zero) is an x value where f(x) = 0, so the curve sits on the x axis. Take
f(x) = x² − x − 6
The graph is a parabola opening upwards that crosses the x axis twice. The grapher lists the roots x = −2 and x = 3. Check by factorising: x² − x − 6 = (x − 3)(x + 2), which is zero exactly when x = 3 or x = −2.
Roots come in two kinds, and the picture tells you which:
- Crossing roots. The curve passes from one side of the axis to the other, so f changes sign. Both roots of x² − x − 6 are like this.
- Touching roots. The curve comes down to the axis and goes back up without crossing. For f(x) = (x − 1)² the only root is x = 1, and f is never negative. Algebraically this is a repeated (double) factor. A sign-change test misses these, which is why the grapher also looks for places where |f| shrinks to a minimum of zero.
The y-intercept
The y-intercept is f(0), the height where the curve crosses the vertical axis. For x² − x − 6 it is (0, −6): put x = 0 and only the constant −6 is left. This is the quickest check that you typed the formula correctly. If the curve crosses the y axis anywhere other than where the constant term says, something in the formula is off.
A function can have no y-intercept: 1/x is not defined at x = 0, so its graph never touches the y axis.
Turning points
A turning point is where the curve stops rising and starts falling (a local maximum) or the reverse (a local minimum). On a parabola there is exactly one, the vertex. For x² − x − 6 the grapher reports min (0.5, −6.25). By hand: the vertex of ax² + bx + c is at x = −b/(2a) = 1/2, and f(0.5) = 0.25 − 0.5 − 6 = −6.25.
A cubic can have two. f(x) = x³ − 3x has a local maximum at (−1, 2) and a local minimum at (1, −2). Its roots are x = 0 and x = ±√3 ≈ ±1.73205. Notice how the roots sit on either side of the turning points: between two neighbouring roots of a smooth function there is always at least one turning point.
"Local" matters. The maximum at (−1, 2) is the highest point near x = −1, not the highest point of the function: x³ − 3x goes up without limit as x grows.
Asymptotes
An asymptote is a line the curve approaches without settling on it. Type
f(x) = (2x + 1)/(x − 3)
and you will see two branches.
- Vertical asymptote at x = 3. The denominator is zero there, so f is undefined, and close to 3 the value shoots towards +∞ on the right and −∞ on the left. The grapher breaks the line at x = 3 instead of drawing a vertical stroke; a vertical line on a graph is almost always a plotting artefact, not part of the function.
- Horizontal asymptote at y = 2. For large x, the +1 and −3 hardly matter and f(x) behaves like 2x/x = 2. Zoom out and the branches flatten towards the line y = 2.
The same function has a root at x = −0.5 (the numerator 2x + 1 is zero) and a y-intercept at (0, −0.333333), that is −1/3. It has no turning points: both branches are falling everywhere.
A sign change across a vertical asymptote is not a root. The function goes from −∞ to +∞ without ever being zero, so the grapher rejects that crossing.
Intersections of two graphs
Where two curves meet, they have the same y for the same x, so the intersection points solve f(x) = g(x). Plot x² − x − 6 together with y = 2x. The grapher gives two intersections:
- (−1.37228, −2.74456)
- (4.37228, 8.74456)
By hand: x² − x − 6 = 2x becomes x² − 3x − 6 = 0, and the quadratic formula gives x = (3 ± √33)/2. Since √33 ≈ 5.74456, the solutions are 4.37228 and −1.37228. The y values are then 2x: 8.74456 and −2.74456. The graph and the algebra agree, which is the point: the picture tells you how many solutions to expect and roughly where, and the algebra gives the exact form. The equation solver shows those steps with the surd kept exact.
A reading checklist
- Scale and grid step.
- Domain: gaps, holes and vertical asymptotes, where f is undefined.
- y-intercept: does it match the constant term?
- Roots: crossing or touching?
- Turning points: how many, maximum or minimum?
- End behaviour: what happens as x goes far left and far right (horizontal asymptote, growth without limit)?
Common mistakes
- Reading a root off the grid by eye. The root of x² − 2 looks like "about 1.4" on a coarse grid; the exact value is √2 ≈ 1.41421. Use the trace or the listed roots, then state the precision you are giving.
- Treating a vertical line as part of the curve. Calculators and spreadsheets often join the two branches of tan(x) or 1/(x − 3) with a vertical stroke. The function has no value there.
- Counting only the roots you can see. Results are for the visible window. x³ − 3x − 18 has a root at x = 3, but a view from −2 to 2 shows none. Zoom out before concluding there are no solutions.
- Mixing degrees and radians. sin(x) on a function graph uses radians, so its roots are at multiples of π ≈ 3.14159, not 180.
- Forgetting the order of operations. −x² is −(x²), an upside-down parabola, and 1/2x is (1/2)·x, a straight line. Brackets decide the shape.
- Calling every high point "the maximum". A local maximum is only the highest point nearby. Say "local" unless you have checked the whole domain.
Once these habits are automatic, a graph becomes a fast way to predict the answer before you calculate it, and to catch a sign error the moment the curve does not match.