What is Graph? — Mathematics

Definition 3.5 High School Mathematics · Chapter 3 — Functions

In a coordinate system, the graph (or curve) of $f$ is the set of all points $(x, f(x))$ for $x$ in the domain. In other words, a point $(x, y)$ lies on the graph exactly when $y = f(x)$ .

Reading an image on a graph: start from x = 3 on the horizontal axis, go vertically to the curve, then horizontally to the vertical axis to read f(3) = 1.25. — Reading an image on a graph: start from $x = 3$ on the horizontal axis, go vertically to the curve, then horizontally to the vertical axis to read $f(3) = 1.25$ .

Solving f(x) = k graphically: the solutions x_1 and x_2 are the abscissas of the intersection points of the curve with the horizontal line y = k. Here f(x) ≤ k holds on [x_1, x_2], where the curve is below the line. — Solving $f(x) = k$ graphically: the solutions $x_1$ and $x_2$ are the abscissas of the intersection points of the curve with the horizontal line $y = k$ . Here $f(x) \leq k$ holds on $\intcc{x_1}{x_2}$ , where the curve is below the line.

Examples

Example 3.7

Is the point $A(2, 5)$ on the graph of $f(x) = x^2 + 1$ ? Compute $f(2) = 4 + 1 = 5$ : yes, since $f(2)$ equals the $y$ -coordinate of $A$ . The point $B(3, 8)$ is not on the graph, because $f(3) = 10 \neq 8$ .

Example 3.10

Consider the function graphed below on $\intcc{-2}{4}$ .

A function defined on [-2, 4], increasing then decreasing. — A function defined on $\intcc{-2}{4}$ , increasing then decreasing.

Reading the graph: $f$ increases from $f(-2) = -1.5$ up to its maximum $f(1) = 3$ , then decreases down to $f(4) = -1.5$ . Its variation table is

$x$	$-2$		$1$		$4$
$f$	$-1.5$	$\nearrow$	$3$	$\searrow$	$-1.5$

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Examples

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