Quadratic Graph Features UPCAT Reviewer: Lesson and Practice

TEACHER ABI UPCAT MATHEMATICS

Quadratic Graph Features

Read the form of the quadratic before doing algebra; each form reveals a different graph feature.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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Quadratic Functions: Vertex and Graph Features

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QUICK REVIEW

Read the form that reveals the feature you need

A quadratic graph is a parabola. Its equation may appear in three useful forms:

Vertex form: y=a(x−h)²+kFactored form: y=a(x−r₁)(x−r₂)Standard form: y=ax²+bx+c

Vertex form reveals the turning point (h,k). Factored form reveals the x-intercepts r₁ and r₂. Standard form reveals the y-intercept c, and its axis is x=−b(2a). The sign of a tells whether the graph opens up or down.

Choose the useful form

Do not expand an equation when its current form already shows the requested feature.

Connect equation and graph

Vertex, intercepts, axis, opening, and range must describe the same parabola.

DO IT FAST

Ask what the current form gives you for free

Problem: Read y=−2(x−3)²+5.

Recognize: It is already in vertex form y=a(x−h)²+k.

Vertex: (3,5)Axis: x=3a=−2, so the graph opens downward

Conclusion: The maximum output is 5, occurring at x=3.

Why it works

The equation’s form encodes graph features. Reading those features directly is faster and less error-prone than expanding every quadratic.

WORKED EXAMPLES

Five forms you should recognize

1. Vertex form

y=2(x−4)²−3

The vertex is (4,−3), the axis is x=4, and the positive coefficient means the graph opens upward. Therefore −3 is the minimum output.

2. Factored form

y=(x+2)(x−5)

Set each factor to zero: x=−2 or x=5. The axis lies halfway between the roots:

x=(−2+5)2=1.5

3. Standard form

y=x²−6x+5

The y-intercept is (0,5). The axis is:

x=−(−6)(2·1)=3

Substituting x=3 gives the vertex (3,−4).

4. Read an actual graph

The graph opens upward, has vertex (2,−1), and crosses the x-axis at 1 and 3. These features match y=(x−2)²−1.

5. Applied maximum

h(t)=−5(t−2)²+23

The negative coefficient means height has a maximum. The vertex shows that the ball reaches 23 m at t=2 s.

COMMON TRAPS

Check before you commit

• Reversing the sign inside x−h
• Writing the axis as y=h
• Confusing the vertex with an intercept
• Calling an upward-opening vertex a maximum
• Expanding when the needed feature is already visible
• Assuming every quadratic crosses the x-axis twice

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

Work at the level you need.

Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

Ready to verify this competency?

A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

QUICK ANSWERS

Quadratic Graph Features FAQ

Why are there two mastery sets?

An unsuccessful attempt loads a different five-question set so memorizing one set does not verify mastery.

Does 5/5 mean permanent mastery?

It verifies the competency today. Revisit it later through mixed practice and simulations.

RELATED COMPETENCIES

Continue your mathematics review.

Quadratic EquationsPolynomial Roots and DiscriminantFunction Transformations

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