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.

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Quadratic Functions: Vertex and Graph Features

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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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Piecewise Functions UPCAT Reviewer: Lesson and Practice

TEACHER ABI UPCAT MATHEMATICS

Piecewise Functions

Choose the interval before calculating; the condition determines which rule is allowed.

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Piecewise Functions

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Choose the condition first; only then use its formula

A piecewise function assigns different formulas to different parts of its domain. For example:

f(x) = x+3   if x<2f(x) = 2x−1   if x≥2

The conditions split the number line at x=2. An input below 2 uses the first formula. An input equal to or above 2 uses the second. The equality sign tells you which branch owns the boundary.

1. Compare the input

Check the input against every condition before calculating.

2. Use one branch only

Once the condition matches, ignore the other formulas.

DO IT FAST

Circle the input, compare it with every condition

Problem: f(x)=x+4 for x<2 and 2x−1 for x≥2. Find f(2).

Recognize: 2 does not satisfy x<2, but it does satisfy x≥2.

Do it fast: Use 2x−1: f(2)=4−1=3.

Why it works

Most errors come from calculating with both formulas or overlooking which inequality contains the equality sign.

WORKED EXAMPLES

Five forms you should recognize

1. Input below the boundary

f(x)={ x+3 if x<2; 2x−1 if x≥2 }

Find f(1). Since 1<2, use x+3:

f(1)=1+3=4Answer: 4

2. Input exactly at the boundary

g(x)={ x+4 if x<3; 2x−1 if x≥3 }

Find g(3). The first condition excludes 3, while x≥3 includes it:

g(3)=2(3)−1=5Answer: 5

3. Open and closed graph endpoints

At x=2, the point at y=4 is open, so it is excluded. The point at y=1 is filled, so it gives the function value.

Therefore f(2)=1.

4. Tiered parking charge

A parking lot charges ₱40 for the first 2 hours and ₱25 for every additional hour.

C(h)={ 40 if 0<h≤2; 40+25(h−2) if h>2 }

For 5 hours, three hours are beyond the first two:

C(5)=40+25(5−2)=40+75=₱115

Do not charge all five hours at ₱25; the first two are already included in ₱40.

5. Making two branches meet

p(x)={ 2x+k if x<3; x²−1 if x≥3 }

To make the graph meet at x=3, the branch values at the boundary must agree. The second branch gives:

3²−1=8

Set the first expression equal to 8:

2(3)+k=8 → 6+k=8 → k=2The branches meet when k=2.

COMMON TRAPS

Check before you commit

• Substituting before checking the condition
• Using both formulas for one input
• Ignoring which inequality contains the equality sign
• Treating an open graph point as included
• Charging every unit at the higher tier instead of only the excess
• Assuming every piecewise graph has a jump

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

Piecewise Functions 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.

FunctionsDomain and RangeLinear Functions

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Mastery results save to your Teacher Abi study profile.

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Domain and Range UPCAT Reviewer: Lesson and Practice

TEACHER ABI UPCAT MATHEMATICS

Domain and Range

Identify possible inputs first, then determine the outputs the function can actually produce.

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Domain and Range

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Domain means allowed inputs; range means resulting outputs

In an ordered pair (x,y), x belongs to the domain and y belongs to the range. Algebraic rules may exclude inputs: a denominator cannot be zero, and an even root must contain a nonnegative value when working with real numbers. Context can restrict inputs further.

Domain

Ask: Which x-values are permitted?

Range

Ask: Which y-values occur after the permitted inputs are used?

DO IT FAST

Input first, restriction second, outputs last

Problem: Find the domain of f(x)=1(x−3).

Recognize: Division by zero is undefined.

Do it fast: x−3=0 at x=3, so exclude 3. Every other real input is allowed.

Why it works

Restrictions come from operations in the rule or limits in the context; they are not arbitrary values to memorize.

WORKED EXAMPLES

Five forms you should recognize

1. Ordered pairs

For (1,4), (2,6), (3,6), the domain is {1,2,3}; the range is {4,6}. Repeated outputs are listed once.

2. Rational rule

For 1(x−5), x=5 is excluded because it makes the denominator zero.

3. Square root

For √(x−2), require x−2≥0, so x≥2.

4. Graph

Project graph points horizontally to read x-values and vertically to read y-values. Open endpoints are not included.

5. Context

If tickets are numbered 1 to 100, the meaningful domain is those whole numbers—not all real numbers.

COMMON TRAPS

Check before you commit

• Switching domain and range
• Listing repeated outputs more than once
• Allowing a zero denominator
• Forgetting square-root restrictions
• Ignoring open endpoints
• Using all real numbers despite a contextual limit

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

Domain and Range 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.

FunctionsPiecewise FunctionsRational Expressions

SAVE AND CONTINUE

Your progress stays on this browser.

Mastery results save to your Teacher Abi study profile.

Return to Student Hub View UPCAT Coverage