Quadratic Forms and Features

Explore standard form, factored form, vertex form.

Lesson 1 of 53 parts Practice Flashcards

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Standard Form

Objective: Explain standard form and use it to solve or interpret a problem.

In y=ax²+bx+c with a≠0, c is the y-intercept and the sign of a determines opening. A contextual x=0 must be meaningful before treating c as a starting value. Set x=0 and evaluate the output.

Worked examples

Try itQuadratic Forms and Features — practice 1.1: For y=(x−4)(x−5), find the y-intercept value.Show answer

Step 1: Set x=0. Step 2: Multiply (−4)(−5). Step 3: The y-intercept value is 20.

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Standard Form

Objective: Explain standard form and use it to solve or interpret a problem.

In y=ax²+bx+c with a≠0, c is the y-intercept and the sign of a determines opening. A contextual x=0 must be meaningful before treating c as a starting value. Set x=0 and evaluate the output.

Worked examples

Try itQuadratic Forms and Features — practice 1.1: For y=(x−4)(x−5), find the y-intercept value.Show answer

Step 1: Set x=0. Step 2: Multiply (−4)(−5). Step 3: The y-intercept value is 20.

Factored Form

Objective: Explain factored form and use it to solve or interpret a problem.

When y=a(x−r₁)(x−r₂), the real roots r₁ and r₂ locate x-intercepts. A repeated factor may touch rather than cross the x-axis. Every point on the x-axis has output zero.

Worked examples

Try itQuadratic Forms and Features — practice 2.1: For y=(x−2)(x−8), what is the larger x-intercept?Show answer

Step 1: Set y=0 to find intercepts. Step 2: Use zero-product: x=2 or x=8. Step 3: The larger is 8.

Vertex Form

Objective: Explain vertex form and use it to solve or interpret a problem.

In y=a(x−h)²+k, the vertex is (h,k). A positive a gives a minimum and a negative a gives a maximum over all real x. The squared factor is zero there.

Worked examples

Try itQuadratic Forms and Features — practice 3.1: For y=3(x−11)²+7, find the vertex y-coordinate.Show answer

Step 1: The squared term is zero at x=11. Step 2: Then y=3(0)+7. Step 3: The vertex y-coordinate is 7.

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