Final Answer:
1. Graph of $ y = x $: a straight line through origin with slope 1.
Graph of $ y = x + 2 $: same line shifted up 2 units.
Graph of $ y = x - 2 $: same line shifted down 2 units.
2. Graph of $ y = x $: as above.
Graph of $ y = x + 2 $: shifted up 2.
Graph of $ y = x - 2 $: shifted down 2.
3. When a number is added or subtracted to the original linear function $ y = x $, the graph shifts
vertically — up if added, down if subtracted. The slope and shape stay the same.
4. Graph of $ y = 2^x $: exponential curve passing through (0,1), rising rapidly for $ x > 0 $, approaching 0 for $ x < 0 $.
Graph of $ y = 2^{x+2} $: same curve shifted
left 2 units.
Graph of $ y = 2^{x-2} $: same curve shifted
right 2 units.
5. When a number is added or subtracted *inside* the exponent (e.g., $ x + c $ or $ x - c $), the graph shifts
horizontally — left if added, right if subtracted. The shape stays the same.
6. Graph of $ y = x $: line through origin, slope 1.
Graph of $ y = 2x $: steeper line through origin, slope 2.
Graph of $ y = \frac{1}{2}x $: flatter line through origin, slope 0.5.
7. Graph of $ y = x $: as above.
Graph of $ y = 2x $: steeper.
Graph of $ y = \frac{1}{2}x $: flatter.
8. When a number multiplies $ x $ in a linear function (e.g., $ y = kx $), the graph’s
steepness changes:
- If $ |k| > 1 $, it gets steeper.
- If $ 0 < |k| < 1 $, it gets flatter.
- Sign of $ k $ determines direction (positive = up-right, negative = down-right).
9. When a number multiplies the entire exponential function (e.g., $ y = a \cdot b^x $):
- If $ a > 1 $, the graph stretches vertically (taller).
- If $ 0 < a < 1 $, it compresses vertically (shorter).
- If $ a < 0 $, it reflects over the x-axis.
The horizontal position and growth rate stay the same.
Parent Tip: Review the logic above to help your child master the concept of linear vs exponential worksheet.
