Graphs, intercepts, and models

Putting equations on a grid

Algebra and geometry joined forces in the 1600s, when René Descartes gave every point in a plane an address \((x, y)\). An equation in \(x\) and \(y\) can then be drawn. A solution point is a pair of numbers that makes the equation true, and the graph of an equation is the set of all its solution points. To sketch one by hand, make a table of solution points, plot them on the grid and join them with a smooth curve. A graph shows every solution at once, which is why graphs reveal what an equation hides.

Plotting too few points can mislead

A quick table is a hypothesis, not a guarantee. Take \(y = x^3 - 12x\) and plot just three points: \((-3, 9)\), \((0, 0)\) and \((3, -9)\). All three lie on a straight line of slope \(-3\), so the sketch suggests a line. The true graph travels much further: at \(x = -2\) it climbs to 16, and at \(x = 2\) it falls to \(-16\), because \((-2)^3 - 12(-2) = -8 + 24 = 16\). Between your plotted points the curve wanders far from any line. Plot enough points to catch the turns.

Intercepts: where a graph meets the axes

Two kinds of solution points guide every sketch. An x-intercept is a point \((a, 0)\) where the graph meets the horizontal axis, and a y-intercept is a point \((0, b)\) where it meets the vertical axis. The zeros do the work: to find the x-intercepts, set \(y = 0\) and solve for \(x\); to find the y-intercept, set \(x = 0\) and solve for \(y\). A graph can have any number of x-intercepts, including none, and graphs that are not functions can even cross the vertical axis more than once. Intercepts are cheap to compute, and they anchor the whole sketch.

Worked example: finding intercepts

Find the intercepts of \(y = x^3 - x^2 - 2x\). For the x-intercepts, set \(y = 0\) and factor:

so \(x = 0\), \(x = 2\) or \(x = -1\), and the x-intercepts are \((-1, 0)\), \((0, 0)\) and \((2, 0)\). For the y-intercept, set \(x = 0\): the equation gives \(y = 0\), so the y-intercept is also \((0, 0)\) — a single point can be both. Checking one point: at \(x = 2\), \(8 - 4 - 4 = 0\), as required.

Symmetry: getting half the sketch free

Symmetry is a pattern in a graph that cuts your work in half. A graph is symmetric with respect to the y-axis when its left half mirrors its right half, with respect to the x-axis when its lower half mirrors its upper half, and with respect to the origin when a half-turn about \((0, 0)\) leaves it unchanged. Algebra tests each one. Replace \(x\) by \(-x\): an equivalent equation means y-axis symmetry. Replace \(y\) by \(-y\): an equivalent equation means x-axis symmetry. Replace both variables: an equivalent equation means origin symmetry. Test before sketching, then plot one half and reflect.

Worked example: testing symmetry

Test \(y = x^3 - 2x\) for symmetry. y-axis: replacing \(x\) by \(-x\) gives \(y = -x^3 + 2x\), which is not an equivalent equation, so there is no y-axis symmetry. Origin: replace both variables to get \(-y = -x^3 + 2x\); multiplying through by \(-1\) returns \(y = x^3 - 2x\), an equivalent equation, so the graph does have origin symmetry. A shortcut is worth knowing: a polynomial in \(x\) with only even powers, such as \(y = x^4 - 3x^2 + 1\), is always symmetric about the y-axis, and one with only odd powers and no constant term is always symmetric about the origin.

Sketching with intercepts and symmetry

Together the tools make short work of an unfamiliar equation. Sketch \(x = y^2 - 4\). Replacing \(y\) by \(-y\) changes nothing, so the graph mirrors across the x-axis: plot the upper half and reflect. Setting \(y = 0\) gives the x-intercept \((-4, 0)\). Two more points above the axis: \(y = 2\) gives \(x = 0\), and \(y = 3\) gives \(x = 5\). So the upper half starts at \((-4, 0)\) and rises through \((0, 2)\) and \((5, 3)\), and reflecting produces a curve opening to the right. The intercept fixed the anchor and symmetry halved the plotting.

Where graphs meet: points of intersection

A point of intersection lies on both graphs at once, so its coordinates satisfy both equations; finding it means solving the equations simultaneously. Where do \(y = x^2 - 1\) and \(y = x + 1\) meet? Set the expressions equal: \(x^2 - 1 = x + 1\), so \(x^2 - x - 2 = 0\), which factors as \((x - 2)(x + 1) = 0\). Then \(x = 2\) or \(x = -1\), giving the points \((2, 3)\) and \((-1, 0)\). Check \((2, 3)\) in both equations: \(2^2 - 1 = 3\) and \(2 + 1 = 3\). In applications, intersections carry meaning: break-even points, crossing times, collisions.

Equations as models

An equation can also stand in for the real world. A mathematical model is an equation built to describe observed data — a growing population, a phone plan's cost, a cooling drink. Building one is a balancing act between two goals that pull in opposite directions: accuracy, so its predictions mean something, and simplicity, so it stays usable. A model also has a domain of times or quantities where it deserves trust, and it describes the trend of the data, never the data point by point.

Worked example: reading a model

A study app counts its active users in thousands and models them with

where \(t\) is years since 2020. The y-intercept is \(N(0) = 50\): the app started with 50 thousand users. At \(t = 4\), the model gives \(N(4) = 8 + 40 + 50 = 98\). When does it reach 200 thousand? Solve \(0.5t^2 + 10t + 50 = 200\). Doubling gives \(t^2 + 20t - 300 = 0\), which factors as \((t + 30)(t - 10) = 0\). The meaningful root is \(t = 10\), the year 2030; the root \(t = -30\) lies outside the model's story. Check: \(N(10) = 50 + 100 + 50 = 200\), and the earlier growth was \(\frac{98 - 50}{4} = 12\) thousand per year against \(\frac{200 - 98}{6} = 17\) later — growth is speeding up.

Common mistakes to avoid

The first slip is swapping the intercept rules: set \(y = 0\) to find x-intercepts, not \(x = 0\). Second, testing symmetry with a single point — the test is an equivalent equation, which speaks for every point on the graph at once. Third, forgetting to replace \(y\) by \(-y\) (not just \(x\) by \(-x\)) when testing for origin symmetry. Fourth, solving an intersection for \(x\) but forgetting to substitute back and recover the \(y\)-coordinates. Fifth, in modelling, quoting a nonsensical root such as \(t = -30\) as an answer, or trusting the model far outside the range of the data behind it.

Key takeaways

The graph of an equation is the set of all its solution points, and a sparse table can misrepresent the shape. Find x-intercepts by setting \(y = 0\), and the y-intercept by setting \(x = 0\). Test symmetry algebraically — replace \(x\) by \(-x\), \(y\) by \(-y\), or both, looking for an equivalent equation — then sketch one half and reflect. Points of intersection satisfy both equations at once; solve the two equations simultaneously. A mathematical model trades accuracy against simplicity: read its y-intercept as the starting value, substitute a target to make predictions, and keep the sensible root.

Question 1 of 5

What are the x-intercepts of the graph of y = x(x - 1)(x + 3)?

Question 2 of 5

What is the y-intercept of the graph of y = x² - 4x + 7?

Question 3 of 5

Which of these graphs is symmetric with respect to the y-axis?

Question 4 of 5

Where do the graphs of y = x - 1 and y = x² - 3x + 2 intersect?

Question 5 of 5

A charity's total donations are modelled by D(t) = 800 + 120t dollars, where t is years since the fund began. What does the y-intercept of this model tell you?