Further Graphs And Tangents
விளையாடிக் கற்றுக்கொள்ளுங்கள்
ஆற்றல் சம்பாதிக்க இந்த கேள்விகளுக்குப் பதிலளியுங்கள், பின்னர் மீன் பிடித்து ஆராயுங்கள். கணக்கு தேவையில்லை.
பாட குறிப்புகள்
Types of Graphs
- Linear: y = mx + c or ax + by = c (straight line).
- Quadratic: y = ax² + bx + c (parabola, symmetric).
- Cubic: y = ax³ + bx² + cx (S-shaped curve).
- Reciprocal: y = a/x + b (two branches, asymptotes at x=0 and y=b).
- Exponential: y = a·kˣ + b (growth if k>1, decay if 0<k<1; asymptote at y=b).
- Trigonometric: sine, cosine, tangent (periodic).
Types of graphs overview
Drawing Graphs from Tables
- Substitute x-values into equation to get y-values; avoid x=0 for reciprocal graphs.
- Plot points accurately (within half a square) and join with a smooth freehand curve.
- Use calculator table function: enter function, start/end x, step size.
- Check for symmetry (e.g., quadratics have vertical line of symmetry).
- Be careful with negative numbers: use brackets and BIDMAS.
A quadratic plotted from a table of values
Solving Equations from Graphs
- Solutions to f(x)=0 are x-intercepts (roots) of the graph.
- To solve f(x)=k, draw horizontal line y=k and read intersection x-coordinates.
- To solve f(x)=g(x), plot both graphs; intersection x-values are solutions.
- Rearrange given equation to match the graph's equation plus a line.
- Only give x-coordinates unless solving simultaneous equations (include y).
Finding the intersection of two graphs
Finding Gradients of Tangents
- Gradient of curve at a point = gradient of tangent at that point.
- Draw tangent by eye using a ruler; extend line for accuracy.
- Calculate gradient = rise/run = Δy/Δx using two far-apart points on tangent.
- Gradient represents rate of change (e.g., speed from distance-time graph).
- Estimation is approximate; exact gradient requires differentiation.
Estimating a gradient from a tangent
Key Features of Reciprocal Graphs
- y = a/x has vertical asymptote at x=0 and horizontal asymptote at y=0.
- y = a/x + b shifts horizontal asymptote to y=b; vertical asymptote remains x=0.
- y = 1/x² is always positive and steeper than y = 1/x.
- No y-intercept or roots for y = a/x.
Asymptotes of y = 1/x
Key Features of Exponential Graphs
- y = kˣ (k>1) shows exponential growth; y-intercept at (0,1); asymptote y=0.
- y = kˣ (0<k<1) shows exponential decay; same intercept and asymptote.
- y = a·kˣ + b has y-intercept at (0, a+b) and horizontal asymptote at y=b.
- Negative powers (e.g., 2⁻ˣ) also represent decay.
Exponential growth
ஸ்லைடுகள்
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பயிற்சி கேள்விகள்
இலவச முன்னோட்டம் — 65-இல் 8 கேள்விகள். அனைத்தையும் பார்க்க பதிவு செய்யவும்.
1.Which of the following is the equation of a reciprocal graph?
Easy- Ay = x2
- By = 1/x
- Cy = 2x
- Dy = x3
2.What is the y-intercept of the graph y = 2x?
Easy- A(0, 0)
- B(0, 1)
- C(0, 2)
- D(1, 2)
3.Which type of graph has a vertical asymptote at x = 0?
Easy- ALinear
- BQuadratic
- CReciprocal
- DExponential
4.The graph of y = x2 is called a
Easy- Astraight line
- Bparabola
- Ccubic curve
- Dhyperbola
5.Which of the following graphs represents exponential decay?
Easy- Ay = 2x
- By = (1/2)x
- Cy = x2
- Dy = 1/x
6.The graph of y = x3 + 2x2 - x - 2 is sketched. How many times does it cross the x-axis?
Medium- A0
- B1
- C2
- D3
7.What is the horizontal asymptote of y = 3/x - 2?
Medium- Ay = 0
- By = -2
- Cx = 0
- Dy = 3
8.The table shows values for y = 2x2 - 3x + 1. Which y-value is missing? | x | 0 | 1 | 2 | |---|---|---|---| | y | 1 | 0 | ? |
Easy- A1
- B3
- C5
- D7
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இந்த தலைப்பிற்கான ஒவ்வொரு கேள்வியையும், ஸ்லைடுகளையும், ஃப்ளாஷ் கார்டுகளையும் மற்றும் திருப்புதல் குறிப்புகளையும் பார்க்க இலவச கணக்கை உருவாக்கவும்.