MATH1202 Chap.4 Parabolas, Hyperbolas and Absolute Value
Parabolas, Hyperbolas and Absolute Value
This chapter compares three graph families with different anchors. Parabolas are organised by a vertex, axis and intercepts; translated hyperbolas by vertical and horizontal asymptotes; absolute-value graphs by reflected portions or corners. These features are then used to solve graphical inequalities as intervals rather than as isolated crossing points.
Several algebraic forms are connected to the feature each exposes most clearly: vertex form for turning behaviour, factor form for zeros and reciprocal form for asymptotes. A disciplined sketch therefore starts with exact features and exclusions, adds a few test points, and only then draws the branches or reflected pieces.
What this chapter covers
- 01
Parabola vertex and axis
- 02
Intercepts and quadratic sign
- 03
Hyperbola centre and asymptotes
- 04
Domain and range exclusions
- 05
Absolute-value reflections
- 06
Graphical inequality intervals
Worked example · free
Solve a quadratic inequality graphically
- +1Factor the boundary equation and find both roots.
- +1Use the leading coefficient to determine opening direction.
- +1Identify where the curve is above the axis.
- +1Include endpoints because equality is permitted.
Key terms
- Vertex
- The turning point anchoring a parabola in vertex form.
- Asymptote
- A line approached by a curve near a boundary or at large magnitude.
- Absolute Value
- A non-negative magnitude that can reflect negative graph portions.
- Boundary Point
- An input where an inequality changes status or becomes undefined.
Parabolas, Hyperbolas and Absolute Value FAQ
What must a parabola sketch show?
Show the vertex, axis of symmetry, opening direction and every intercept requested by the question. These features make later conclusions about range, maxima, minima and inequalities visible rather than implied.
Can a graph cross an asymptote?
For the translated reciprocal functions taught here, the vertical asymptote is excluded from the domain and the horizontal asymptote is excluded from the range. Build the sketch from these exclusions and test points on both sides.
How do I solve an inequality from a graph?
Find equality boundaries, divide the number line into intervals, and determine where the curve has the required sign or relative position. Include a boundary only when equality is allowed and the function is defined there.
Which form of a quadratic should I use?
Use factor form when roots and sign intervals are central, vertex form when the turning point and range are central, and expanded form when the vertical intercept or coefficient comparison is useful. Convert only when the target feature is hidden. Whichever form you choose, the leading coefficient must give the same opening direction and vertical scale.
How do transformations move reciprocal asymptotes?
In a translated reciprocal rule, the input shift locates the vertical asymptote and the output shift locates the horizontal asymptote. Scale and reflection change branch orientation but not those asymptote positions. State excluded input and output values explicitly, then use one test point in each domain interval to place the branches correctly.
When should an absolute-value rule be split into cases?
Split where the expression inside the absolute value equals zero. On each resulting interval, decide whether that expression is non-negative or negative and replace the absolute value accordingly. Solve or sketch each valid piece only on its own interval. This prevents a correct local formula from being applied beyond the point where its sign changes.
How can intercepts expose a poor sketch?
Substitute zero for x to obtain the vertical intercept and solve the rule equal to zero for horizontal intercepts. These exact points must lie on the finished curve and agree with the claimed domain. If a branch crosses an excluded line, a parabola has the wrong number of roots, or a corner misses its calculated location, revise the sketch.
What should an interval answer show?
An inequality answer should identify every boundary, indicate whether each boundary is included, and exclude points where the function is undefined. Use a sign chart or labelled graph to justify which intervals satisfy the comparison. Then write the result in consistent interval or inequality notation. Checking one test value from every interval prevents selecting a region only because it looks plausible on a rough sketch.
Exam move
Memorise the defining anchor for each family, then calculate features before drawing. Build a comparison grid listing formula form, anchor, exclusions, intercept method and end behaviour. Use a sign chart beside every graphical inequality and test one value from every interval. For parabolas, convert among expanded, factor and vertex forms, connecting the leading coefficient to opening and range.
For hyperbolas, state both exclusions and place a test point on each side of the vertical asymptote. For absolute value, decide whether the output or input is being reflected and locate every corner exactly. Practise reconstructing a plausible rule from a labelled graph as well as sketching from a rule. Check the finished sketch against the algebraic form, intercepts, domain and range.
Working through Parabolas, Hyperbolas and Absolute Value in MATH1202? Sia is AskSia’s AI Mathematics tutor — ask any MATH1202 Parabolas, Hyperbolas and Absolute Value question and get a clear, step-by-step explanation grounded in how MATH1202 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
