Question 1Mid
The domain of is all real such that . Let . Which of the following inequalities describes the domain of ?
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Answer: D —
The input to must satisfy . Subtracting 6 gives , which is the domain of .
Study on the flyACTMathematicsKey features and transformations of graphs
Mathematics
Key features and transformations of graphs — ACT practice questions
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What this topic is
Shifts, stretches and reflections make up the transformation half of Key features and transformations of graphs.
Replacing x with x plus three moves a graph three units left, not right, which is the most frequent mix-up. Multiplying the whole function by a number stretches it vertically. A horizontal shift also carries a domain boundary along: if f is defined for x at least zero, then f of x plus three is defined for x at least negative three.
Among the key features are intercepts, slope, maximum and minimum points and asymptotes; ACT Mathematics asks for an intercept, a stretch factor or a shifted domain.
Sample questions
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Question 2Easier
The graph of is a vertical stretch of the graph of , where and . If and , what is the value of ?
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Answer: B — 2.5
Since , the given values yield . Thus, .
Question 3Easier
In the standard coordinate plane, a line has slope and -intercept 6. What is the -intercept of this line?
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Answer: C — 3
The equation is . Setting gives . The number 6 is the -intercept.
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More ACT topics
Mathematics
- Real number system and number properties
- Integer and rational exponents
- Radicals and roots
- Complex numbers
- Vectors and matrices
- Fractions, decimals, and operations
- Percents and percent change
- Ratios, rates, and proportional relationships
- Linear equations and expressions in one variable
- Inequalities (linear and compound)
- Systems of equations
- Polynomial expressions and factoring
- Quadratic equations
- Radical and rational equations
- Exponential relationships
- Function notation and evaluating functions
- Linear functions and graphs
- Quadratic and polynomial functions
- Radical and rational functions
- Piecewise functions
- Exponential and logarithmic functions
- Lines, angles, and polygons
- Triangles — congruence, similarity, Pythagorean theorem
- Circles — arcs, sectors, angles, equations
- Surface area and volume of solids
- Coordinate geometry — distance, midpoint, slope, line equations
- Conic sections
- Right-triangle trigonometry and trig functions
- Measures of center and spread
- Data collection methods and study design
- Bivariate data and relationships (scatterplots, correlation)
- Probability and sample spaces
- Counting, combinations, and permutations
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