Answer:
1. the graph is reflected in the x-axis, and translated 2 units right and 3 units down ⇒ answer D
2. His average running speed is 10 km/h ⇒ answer A
3. There will be 21.9 days before only 10 g remain ⇒ answer B
Explanation:
* Lets solve the problems
1.
- The quadratic function in the form y = a(x - p)² + q represented by
upward parabola with minimum vertex (p , q) if a is a positive value
∵ The sign of a is changed to opposite
∴ The parabola will opened downward with maximum vertex
- That means the parabola reflected across the x-axis
∵ The coordinates of the vertex are (p , q)
∵ p is added by 2 and q is decreased by 3
∴ The new vertex is (p + 2 , q - 3)
- That means the parabola will translated 2 units
to the right and 3 units down
∴ The graph of the quadratic function y = a(x - p)² + q is reflected
in the x-axis, and translated 2 units right and 3 units down.
2.
- On average, Jesse can bike five times as fast as he can run
* Let his average speed of run is x
∵ His average speed of run is x
∴ His average speed of bike is 5x
- To travel 25 km, he needs 2 h more if he is running than if he
is biking
∵ The time = distance/speed
∵ The distance is 25 km
∵ His running speed is x
∴ His time of running = 25/x
∵ His biking speed is 5
∴ His time of biking = 25/5x ⇒ divide up and down by 5 to simplify
∴ His time of biking = 5/x
∵ He needs 2 h more if he is running than if he is biking
- That means the time of running = 2 + the time of biking
∴ 25/x = 2 + 5/x
- Multiply all terms by x
∴ 25 = 2x + 5
- Subtract 5 from both sides
∴ 20 = 2x ⇒ divide the two sides by 2
∴ x = 10
∵ x is his average running speed
∴ His average running speed is 10 km/h
3.
- The half-life of an isotope is 5.6 days
∴ The time of every half-life is 5.6
- There is originally 150 g of the isotope
* Consider that its wight decreased to half every 5.6 days
∴ W = a(b)^t , where a is the original value and b is the factor of
decreasing and t is the number of the 5.6 days
∵ a = 150 , W = 10 , b = 0.5
∴ 10 = 150(0.5)^t
- divide both sides by 150
∴ 10/150 = (0.5)^t ⇒ simplify it
∴ 1/15 = (0.5)^t
- Insert ㏑ for both sides
∴ ㏑(1/15) = ㏑(0.5)^t
- Remember ㏑(a)^m = m ㏑(a)
∴ ㏑(1/15) = t㏑(0.5)
- Divide both sides by ㏑(0.5)
∴ t = ㏑(1/15) ÷ ㏑(0.5) = 3.90689
∵ t is the number of the 5.6 days
- To find the number of the days multiply t by 5.6
∴ The number of the days = 3.90689 × 5.6 = 21.87858 ≅ 21.9
∴ There will be 21.9 days before only 10 g remain