X-Intercepts & Writing Quadratic Equations

Find roots and write quadratic equations from conditions. Grade 11 Math 20-1 Alberta diploma prep.

Lesson 1.5 of Quadratic Functions & Equations in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Number of X-Intercepts. Determine whether a parabola has zero, one, or two x-intercepts by comparing the sign of a and the sign of q in vertex form.
• Writing Quadratic Equations from Information. Use the vertex, another point, or given conditions to find a, p, and q and write a complete vertex form equation.
• Tracking Points and Symmetry. Apply transformations to any point on y = x² and use equal y-values to locate the axis of symmetry.
• Increasing and Decreasing Intervals. Identify where a parabola is increasing or decreasing, and solve a multi-condition challenge using a system of equations.

Inside the lesson — a free preview

The Three Cases

A parabola can have zero, one, or two x-intercepts. Which case occurs depends on two things: the direction of opening (sign of a) and the location of the vertex (value of q).

The Conditions

For a quadratic in vertex form y = a(x - p)^2 + q, the number of x-intercepts is determined entirely by the signs of a and q.

The Key Insight. Compare the sign of a (which way the parabola opens) with the sign of q (where the vertex sits vertically).

Same signs — the vertex is on the same side of the x-axis as the direction of opening, so the parabola never crosses.

Opposite signs — the vertex is on the opposite side, so the parabola must cross twice.

q = 0 — the vertex sits right on the x-axis, so the parabola touches it exactly once.

Worked example: Predict the Number of X-Intercepts

For each equation, identify a and q, compare their signs, and state the number of x-intercepts.

Part a) — y = -5(x - 3)^2 - 2. Identify parameters: a = -5, q = -2.

Compare signs: both are negative — same sign.

Conclusion: zero x-intercepts.

Visualize it: the parabola opens down with vertex at (3, -2), which is below the x-axis. Opening down means it travels further below — it never reaches the x-axis.

Part b) — y = 2(x + 5)^2. Identify parameters: a = 2, q = 0.

The condition q = 0 is met.

Conclusion: one x-intercept.

Visualize it: the vertex is at (-5, 0), sitting right on the x-axis. The parabola opens up and just touches the axis at that point.

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