11.3 Graphs, Equations and Inequality Regions

Quick Answer: An equation is a graph: the set of points that make it true. An inequality is a region: all points that make it true, usually a half-plane or the inside/outside of a curve. A system is the overlap. Shade after you decide solid versus dashed, then test one point.

Current nbt.ac.za Test Content asks you to identify relationships between graphs and their equations, or inequalities and the regions they describe. That is a matching skill as much as an algebra skill. Independent OpenExamPrep examples below use original lines and curves.

Graphs and equations

The graph of y = f(x) is every point (x, f(x)). Asking "which equation matches this sketch?" is the same as asking which rule those points obey.

A point on a graph

(3, 7) lies on y = 2x + 1 because 7 = 2(3) + 1. It does not lie on y = 2x − 1, because 7 ≠ 5. You can eliminate options on MAT by testing one clearly marked point — often an intercept — before you solve anything lengthy.

Intercepts as solutions

• y-intercept: the solution of the equation together with x = 0.
• x-intercept: the solution of the equation together with y = 0, equivalently the roots of f(x) = 0.

For y = x² − 8x + 12, the x-intercepts 2 and 6 are the solutions of x² − 8x + 12 = 0. The picture and the algebra are the same fact.

Intersection of two graphs

Where y = f(x) and y = g(x) cross, f(x) = g(x). That x, with either formula for y, is the intersection point.

Worked example: y = 2x + 1 and y = −x + 7.

2x + 1 = −x + 7 → 3x = 6 → x = 2 → y = 5. Intersection (2, 5).

Check on both: 2(2)+1 = 5 and −2+7 = 5. If an option says (5, 2), it swapped coordinates — a common trap when someone solves for x and then stops thinking.

A line and a parabola can meet twice, once, or not at all. Example: y = x + 2 and y = x² − 4x + 6.

x + 2 = x² − 4x + 6 → 0 = x² − 5x + 4 = (x − 1)(x − 4). Meetings at x = 1 and x = 4, points (1, 3) and (4, 6). Two solutions mean two intersection points on the sketch.

If you are given a labelled sketch instead of equations, read the crossing first, then see which pair of equations is satisfied there. The relationship the Test Content names is exactly this: graph ↔ equation.

Inequalities and the regions they describe

Replace "=" with an inequality and the graph becomes a boundary of a region.

Lines and half-planes

For a line y = mx + c:

"Above" means larger y, toward the top of the usual MAT sketch. It is not automatically "to the right." For a steep vertical-looking line, still compare y to the line's y-value at that x, or rewrite as x < something if the question used a vertical boundary.

Vertical line: x = k. Then x ≥ k is the closed half-plane to the right, including the line. x < k is the open half-plane to the left, dashed line.

Horizontal line: y = k. Then y ≤ k is on or below the line.

Worked example: y ≤ −x + 4.

• Boundary is the line y = −x + 4, drawn solid because equality is allowed.
• Shade the half-plane on or below the line (smaller y).
• Test (0, 0): 0 ≤ 4, true, so the origin is in the region. Test (0, 5): 5 ≤ 4, false, so above the line is out.

If the inequality had been y < −x + 4, the same side would be shaded but the line itself would be dashed and points on the line would fail.

Curves: below a parabola, outside a hyperbola

The same y-comparison works for a function graph. y ≤ x² − 6x + 5 means, at each x, the point sits on or below the parabola. Because this parabola opens up, "below" includes the interior of the U-shape.

For a hyperbola y = 6/x, the plane is cut by the asymptotes as well as the branches. Testing a point is safer than memorising a picture: (2, 1) satisfies 1 ≤ 6/2, while (2, 5) does not.

Circle inequalities appear because the booklet also lists circle graphs. (x − 1)² + (y + 2)² ≤ 9 is the disk of radius 3 about (1, −2), boundary included. Strict < is the open disk, dashed circle.

Systems: shading, intersection, and the winning region

A system of inequalities is true only where every inequality is true. On a MAT sketch with regions labelled P, Q, R, S, you are looking for the overlap — the intersection of half-planes (and maybe a curved side).

Worked system

x ≥ 0, y ≥ 0, x + y ≤ 6

• x ≥ 0: on or to the right of the y-axis
• y ≥ 0: on or above the x-axis
• x + y ≤ 6, or y ≤ −x + 6: on or below the line through (6, 0) and (0, 6)

The overlap is the right triangle with vertices (0, 0), (6, 0) and (0, 6), including the three sides. A point such as (2, 2) is inside: 2+2 ≤ 6. A point such as (5, 4) is outside: 9 ≤ 6 is false. A point such as (3, −1) fails y ≥ 0 even though 3−1 ≤ 6.

If the third inequality were x + y < 6, the hypotenuse would be dashed and (6, 0) would not belong.

A mixed line-and-curve system

y ≥ x and y ≤ 8 − x², with no extra restriction.

• y ≥ x: on or above the line y = x
• y ≤ 8 − x²: on or below the downward parabola y = 8 − x² (vertex (0, 8))

They meet where x = 8 − x² → x² + x − 8 = 0. You may not need the exact roots if the sketch already marks the two crossing points; you only need to recognise that the feasible set is the finite cap between the line and the parabola, not the infinite half-plane above the line.

Procedure when the question shows four shaded labels

1. Mark each boundary as solid or dashed from the inequality symbols.
2. For each boundary, name which side is allowed (test (0, 0) if it is not on a boundary; otherwise test (1, 0) or (0, 1)).
3. The correct region is the one that sits on the allowed side of every boundary.
4. If two regions look similar, use a point that is in one and not the other, and test the full system.

Example: y ≥ 2, x ≤ 3, y ≤ x. The origin fails y ≥ 2; (4, 3) fails x ≤ 3; (2.5, 3) fails y ≤ x. The point (2.5, 2.2) satisfies all three. That is how adjacent labels separate without shading the whole page.

Equations as the "skin" of a region

Sometimes one option is the boundary equation rather than the region. y = −x + 4 is only the line. y ≤ −x + 4 is the line and the half-plane on or below it. MAT will offer both. The Test Content wording — inequalities and the regions they describe — is a hint that the answer is usually an area, not only a curve, unless the item asks for the intersection of two graphs (a point or a pair of points).

Independent OpenExamPrep habit: write the system next to the sketch, tick each inequality for one interior point, and only then choose the label. No calculator is involved; the arithmetic is comparison, not decimal graphing.