OMPT drill — Functions

OMPT linear and quadratic functions practice: lesson and 10 problems

Slopes, intercepts, vertices, and intersections. If you can find where a line meets a parabola quickly, a surprising share of the test opens up. Attempt each problem before opening the solution; the notation only becomes automatic by writing it yourself.

Tested inOMPT-A OMPT-B OMPT-C OMPT-D OMPT-E OMPT-F OMPT-G

OMPT linear and quadratic functions: the lesson

What the OMPT tests in linear and quadratic functions

A linear function $f (x) = a x + b$ draws a straight line. The slope $a$ is how much $y$ changes when $x$ increases by one, and $b$ is where the line crosses the $y$ -axis. A quadratic function $f (x) = a x^{2} + b x + c$ draws a parabola, opening upward when $a > 0$ and downward when $a < 0$ , with everything interesting about it concentrated in one point, the vertex at $x = - \frac{b}{2 a}$ . These two shapes appear on every variant from OMPT-A to OMPT-G; on OMPT-C only the line has its own chapter.

The OMPT rarely asks about a shape in isolation. The productive question is where things meet: where a line crosses the axes, where two lines cross, where a line meets a parabola, and how many times. Each of those reduces to the same move, set two expressions equal and solve. The heavy sub-skills are building a line from two points, locating a vertex, and intersecting. Perpendicular slopes, a tangent fixed by a parameter, and a short optimisation story make up the rest. Questions are phrased as "give the formula of the line", "find the vertex", "for which value of $c$ ", or "how many points of intersection".

Notation and answer format the OMPT expects

A line is typed as a formula, $y = 3 x - 1$ , with slope and intercept exact: a slope of $- \frac{1}{2}$ is typed as $- 1/2$ , not $- 0.5$ , unless the question says "round to".
Points are pairs with the $x$ -coordinate first: $(2, 3)$ . An intersection question wants points, not just $x$ -values.
A vertex is a point too, $(2, - 3)$ . "The minimum" means the $y$ -value; "where the minimum is" means the $x$ -value.
A slope is a number, possibly a fraction, possibly negative. A horizontal line has slope 0; a vertical line has no slope and is not a function.
A "how many intersections" question wants 0, 1 or 2; a "for which $c$ " question wants the value of the parameter.

Worked example 1: a line through two points

Find the line through $(1, 2)$ and $(3, 8)$ .
Slope first: $a = \frac{8 - 2}{3 - 1} = \frac{6}{2} = 3$ . Rise over run, and keep the two points in the same order top and bottom.
Feed one point into $y = 3 x + b$ . Using $(1, 2)$ : $2 = 3 + b$ , so $b = - 1$ .
The line is $y = 3 x - 1$ .
Check with the point you did not use: at $x = 3$ , $y = 9 - 1 = 8$ . Correct. That single check catches slope and intercept errors at once.

Worked example 2: vertex and minimum value

Find the minimum of $f (x) = x^{2} - 4 x + 1$ and say where it occurs.
The parabola opens upward ( $a = 1 > 0$ ), so the vertex is the minimum.
Vertex $x$ -coordinate: $x = - \frac{- 4}{2 \cdot 1} = 2$ . Two minus signs make a plus.
Height: $f (2) = 4 - 8 + 1 = - 3$ . The minimum value is $- 3$ , reached at $x = 2$ .
Reread the question. "The minimum" is $- 3$ ; "where" is $x = 2$ ; "the vertex" is $(2, - 3)$ . Hand over the one asked for.

The full lesson adds perpendicular lines, the three forms of a parabola and how to move between them, building a parabola from a vertex and a point, the discriminant as a count of intersections, a tangent line fixed by a parameter, and a revenue word problem. It ends with the edge cases the OMPT likes and the slips I mark most often.

How each OMPT variant tests linear and quadratic functions

The chapter weights are the official ones from the omptest.org learning-assessment pages. The question counts are my estimates of how much of that chapter this one topic takes up. The 21-question papers (OMPT-D and OMPT-F) ask long multi-part questions, so a topic there usually shows up as a step inside a bigger problem.

OMPT-A
52 questions
Testedroughly 5 to 7 of 52 questions
Official chapter: Quadratic equations and functions, 22% of the paper.
Lines through two points, a vertex or minimum, intersections of a line and a parabola, and one "for which value of the parameter" item. Answers are formulas, points or single numbers.
OMPT-B
64 questions
Testedroughly 5 to 7 of 64 questions
Official chapter: Quadratic equations and functions, 10% of the paper.
Same shapes as OMPT-A, plus tangent questions via the discriminant that later reappear with derivatives. Perpendicular lines and "how many intersections" are regular.
OMPT-C
30 questions
Testedroughly 2 to 4 of 30 questions
Official chapter: Linear formulas and equations, 25% of the paper.
Mostly lines, through the Linear formulas and equations chapter: slope, intercepts, the formula through two points, and a graph to read. Parabola items are light, a vertex or an intercept at most.
OMPT-D
21 questions
Tested1 to 2 of the 21 long questions, usually as a step inside a bigger problem
Official chapter: Functions, equations and inequalities, 30% of the paper.
A line or parabola opens a long question that goes on to tangents, areas or trigonometry. Expect a parameter and a "for which values" part.
OMPT-E
40 questions
Testedroughly 3 to 5 of 40 questions
Official chapter: Quadratic equations and functions, 10% of the paper.
Lines from data (cost per unit, fixed charge), vertex of a parabola in a story, and line-parabola intersections. Numeric answers, occasional graph reading.
OMPT-F
21 questions
Tested1 to 2 of the 21 long questions, usually as a step inside a bigger problem
Official chapter: Functions, 15% of the paper.
Business framing: linear cost and demand, quadratic revenue or profit, the price that maximises it, break-even as an intersection. Multi-part, with a parameter in the later parts.
OMPT-G
26 questions
Testedroughly 2 to 4 of 26 questions
Official chapter: Functions, graphs and equations, 45% of the paper.
Lines and parabolas arrive through mechanics: distance-time lines, a height-time parabola, when a projectile lands or peaks. Give the number with its unit.

OMPT linear and quadratic functions practice problems

10 original problems in the formats the OMPT uses: typed numeric answers, multiple choice, and spot-the-error. The last few are harder than the exam average on purpose. Solutions and the common mistake sit behind each problem.

Problem 1Numeric answer

A line passes through the points $(1, 3)$ and $(4, 12)$ . What is its slope?

Show the worked solution

Answer: 3

Slope is rise over run: $\frac{12 - 3}{4 - 1} = \frac{9}{3} = 3$ . I keep the subtraction order consistent — both differences go second point minus first point.

Problem 2Multiple choice

What is the vertex of the parabola $y = x^{2} - 6 x + 5$ ?

A $(3, - 4)$
B $(- 3, - 4)$
C $(3, 4)$
D $(6, 5)$

Show the worked solution

Answer: A — $(3, - 4)$

The vertex sits at $x = - \frac{b}{2 a} = - \frac{- 6}{2} = 3$ . Substituting back: $y = 9 - 18 + 5 = - 4$ . So the vertex is $(3, - 4)$ . Completing the square tells the same story: $y = (x - 3)^{2} - 4$ .

Problem 3Numeric answer

A line has slope $- 2$ and passes through $(3, 1)$ . What is its $y$ -intercept?

Show the worked solution

Answer: 7

I write $y = - 2 x + b$ and force the line through $(3, 1)$ : $1 = - 2 \cdot 3 + b$ , so $1 = - 6 + b$ and $b = 7$ . The line is $y = - 2 x + 7$ , which indeed hits $(0, 7)$ on the $y$ -axis.

Problem 4Multiple choice

The parabola $y = - 2 (x + 1)^{2} + 8$ crosses the $x$ -axis at two points. What is the sum of their $x$ -coordinates?

A $- 2$
B $2$
C $4$
D $- 3$

Show the worked solution

Answer: A — $- 2$

Setting $y = 0$ : $- 2 (x + 1)^{2} + 8 = 0$ gives $(x + 1)^{2} = 4$ , so $x + 1 = \pm 2$ and $x = 1$ or $x = - 3$ . The sum is $1 + (- 3) = - 2$ . Symmetry offers a shortcut: the intercepts straddle the axis of symmetry $x = - 1$ , so their sum must be $2 \cdot (- 1) = - 2$ .

Problem 5Spot the error

A student finds the $x$ -intercept of the line $y = 3 x - 12$ . Step 1: set $x = 0$ . Step 2: then $y = - 12$ . Step 3: so the $x$ -intercept is $(0, - 12)$ . Which step contains the error?

Show the worked solution

Answer: Step 1

The $x$ -intercept is where the line crosses the $x$ -axis, which means $y = 0$ , not $x = 0$ . Setting $y = 0$ : $0 = 3 x - 12$ , so $x = 4$ , and the intercept is $(4, 0)$ . What the student actually computed in Steps 2 and 3 is the $y$ -intercept, $(0, - 12)$ .

Problem 6Numeric answer

A ball is thrown upward and its height in metres after $t$ seconds is $h (t) = - 5 t^{2} + 20 t$ . After how many seconds does it reach its highest point?

Show the worked solution

Answer: 2

The height function is a downward parabola, so the maximum is at the vertex: $t = - \frac{b}{2 a} = - \frac{20}{2 \cdot ( - 5 )} = 2$ seconds. At that moment the height is $h (2) = - 20 + 40 = 20$ metres, but the question only asks for the time.

Problem 7Multiple choice

Which line passes through $(- 2, 4)$ and is parallel to $y = - 3 x + 1$ ?

A $y = - 3 x + 10$
B $y = - 3 x - 2$
C $y = 3 x + 10$
D $y = \frac{1}{3} x + \frac{14}{3}$

Show the worked solution

Answer: B — $y = - 3 x - 2$

Parallel means the same slope, so the line is $y = - 3 x + b$ . Forcing it through $(- 2, 4)$ : $4 = - 3 \cdot (- 2) + b = 6 + b$ , so $b = - 2$ . The line is $y = - 3 x - 2$ . Check: at $x = - 2$ , $y = 6 - 2 = 4$ . The option with $+ 10$ comes from writing $- 3 \cdot (- 2)$ as $- 6$ ; the one with slope $\frac{1}{3}$ is the perpendicular line, not the parallel one.

Problem 8Numeric answer

The parabola $y = x^{2} + b x + 12$ has its vertex at $x = 4$ . What is $b$ ?

Show the worked solution

Answer: -8

The vertex of $a x^{2} + b x + c$ sits at $x = - \frac{b}{2 a}$ . With $a = 1$ that is $- \frac{b}{2} = 4$ , so $b = - 8$ . The parabola is $y = x^{2} - 8 x + 12$ . Check by completing the square: $(x - 4)^{2} - 4$ , vertex at $(4, - 4)$ , and the axis is indeed $x = 4$ .

Problem 9Numeric answer

For which value of $k$ is the line $y = x + k$ tangent to the parabola $y = x^{2} - 3 x + 5$ ?

Show the worked solution

Answer: 1

Set the formulas equal: $x^{2} - 3 x + 5 = x + k$ , so $x^{2} - 4 x + (5 - k) = 0$ . Tangent means exactly one solution, so the discriminant is zero: $D = 16 - 4 (5 - k) = 16 - 20 + 4 k = 4 k - 4$ , and $4 k - 4 = 0$ gives $k = 1$ . Check the contact point: with $k = 1$ the quadratic is $x^{2} - 4 x + 4 = (x - 2)^{2}$ , so $x = 2$ ; the parabola gives $4 - 6 + 5 = 3$ and the line gives $2 + 1 = 3$ . They touch at $(2, 3)$ .

Problem 10Numeric answer

A farmer has 60 metres of fence to enclose a rectangular pen against a long straight wall. The wall forms one side, so the fence covers only the other three sides. What is the largest area, in square metres, the pen can have?

Show the worked solution

Answer: 450

Let $x$ be the length of each side perpendicular to the wall. Two of those use $2 x$ metres, leaving $60 - 2 x$ for the side parallel to the wall. Area: $A (x) = x (60 - 2 x) = 60 x - 2 x^{2}$ , a downward parabola. Its maximum is at the vertex, $x = - \frac{60}{2 \cdot ( - 2 )} = 15$ . Then the long side is $60 - 30 = 30$ and the area is $15 \cdot 30 = 450$ square metres. Check a neighbour: $x = 14$ gives $14 \cdot 32 = 448$ , smaller, as it should be.

OMPT linear and quadratic functions: questions students ask

Are linear and quadratic functions on the OMPT-A?

Yes. They span two OMPT-A chapters, Linear formulas and equations and Functions, with Quadratic equations feeding the parabola work, so roughly 5 to 7 of the 52 questions come from here. Expect to build a line from two points, find a vertex or a minimum, intersect a line with a parabola, and answer one question with a parameter in it.

How do I find the vertex of a parabola for the OMPT?

For y = ax squared + bx + c the vertex is at x = -b over 2a; substitute that x back in to get the height. If the parabola is given in factored form, the vertex sits halfway between the two x-intercepts. If it is in vertex form a(x - p) squared + q, the vertex is (p, q) and there is nothing to compute. Watch the sign when b is already negative: two minuses make a plus.

How do I find where a line meets a parabola on the OMPT?

Set the two formulas equal, move everything to one side so a quadratic equals zero, and solve it. Then substitute each x back into the line to get the y-values, because the question wants points. The discriminant tells you how many meetings there are before you solve: two if positive, one if zero (tangent), none if negative. "For which c is the line tangent" means "for which c is the discriminant zero".

What is the difference between a linear function and a linear equation on the OMPT?

A linear equation such as 3x - 12 = 0 has one unknown and one answer, x = 4. A linear function y = 3x - 12 pairs every input with an output and draws a line; questions ask for its slope, its intercepts or where it meets another graph. Solving the equation is the same as finding the x-intercept of the function, which is why the two topics sit next to each other on every variant.

Is the parabola on the OMPT-C?

Only lightly. The OMPT-C syllabus has Numbers, Algebra, Linear formulas and equations and Geometry, so lines carry the weight: slope, intercepts, a formula through two points and reading a graph. A quadratic can appear through the Algebra chapter, usually as an expansion or a factorisation, and at most a simple vertex or intercept. Spend OMPT-C time on lines and geometry.

Do I need a calculator for line and parabola questions on the OMPT?

No. The maths variants provide no calculator unless a question supplies one, and the numbers are chosen so that slopes are small fractions and vertices land on integers or halves. If a vertex computation gives an ugly decimal, recheck -b over 2a, in particular the sign of b. Type exact values: a slope of -1/2 is typed as -1/2, not -0.5.

Test hubs that cover linear and quadratic functions:OMPT-A OMPT-B OMPT-C OMPT-D OMPT-E OMPT-F OMPT-G All 28 topics

OMPT linear and quadratic functions practice: lesson and 10 problems

OMPT linear and quadratic functions: the lesson

What the OMPT tests in linear and quadratic functions

Notation and answer format the OMPT expects

Worked example 1: a line through two points

Worked example 2: vertex and minimum value

How each OMPT variant tests linear and quadratic functions

OMPT linear and quadratic functions practice problems

OMPT linear and quadratic functions: questions students ask

Related OMPT topics and test hubs