Subscribe

9th grade · Math · Unit 20 of 22

Parabolas & Lines

Forms of a quadratic and where a line meets it

What your child practices

Rewriting a quadratic in factored form to show its zeros and in vertex form to show its vertex, and solving a system of a line and a parabola by setting them equal and factoring.

New York State Learning Standards learning standard: AI-F.IF.8; AI-A.REI.7 · writing quadratic functions in equivalent forms to show zeros and the vertex, and solving a system of a linear and a quadratic equation

How we explain it

Children can open this short lesson before they practice.

  1. Factored form y = (x − r)(x − s) shows the zeros r and s.
  2. Vertex form y = (x − h)² + k shows the vertex (h, k).
  3. To solve a line and a parabola together, set the two expressions for y equal and solve the quadratic.

Example: Where do y = x² and y = x + 2 meet? x² = x + 2, so x² − x − 2 = 0. (x − 2)(x + 1) = 0, so x = 2 or x = −1. y = 4 or y = 1. Answer: (2, 4) and (−1, 1)

Parabolas & Lines in New York

In the New York State Learning Standards, this is unit 20 of 22 in 9th grade Math, after Comparing Functions and before Units & Accuracy.

Gradelings describes a child's practice on this unit as Beginning, Approaching, Meeting or Exceeding (Four practice steps). That reflects practice, not a report-card mark: the teacher decides the mark.

Source: New York State Education Department (NYSED)

Sample questions

A taste of this unit. In the app, questions change every time and get easier or harder to match your child.

  1. Question 1

    What are the zeros of y = (x + 5)(x − 4)?

    • x = −20
    • x = −1
    • x = −4 and x = 5
    • x = −5 and x = 4

  2. Question 2

    Which is y = x² − 4x + 5 written in vertex form, to show its vertex?

    • y = (x + 2)² + 1
    • y = (x − 4)² + 1
    • y = (x − 2)² + 1
    • y = (x − 2)² + 5

  3. Question 3

    Solve the system y = x² + 3 and y = −9x − 17. What are the x-values where they meet?

    • x = −9 and x = −17
    • x = 4 and x = 5
    • x = −5 and x = −4
    • x = −5 only

  4. Question 4

    What is the vertex of y = (x − 1)² + 1?

    • (−1, 1)
    • (1, −1)
    • (1, 1)

  5. Question 5

    At most how many points can a line and a parabola share?

    • infinitely many
    • 3
    • 1
    • 2

  6. Question 6

    Solve the system y = x² − 3 and y = 3x − 5. What are the x-values where they meet?

    • x = 1 only
    • x = 3 and x = −5
    • x = 1 and x = 2
    • x = −2 and x = −1

More 9th grade