1

Section 1.4 Formulas for Linear Functions 2

A grapefruit is thrown into the air. Its velocity, v, is a linear function of t, the time since it was thrown. (A positive velocity indicates the grapefruit is rising and a negative velocity indicates it is falling.) Check that the data in Table 1.30 corresponds to a linear function. Find a formula for v in terms of t Page 27 Example 1

A grapefruit is thrown into the air. Its velocity, v, is a linear function of t, the time since it was thrown. (A positive velocity indicates the grapefruit is rising and a negative velocity indicates it is falling.) Check that the data in Table 1.30 corresponds to a linear function. Find a formula for v in terms of t.1.30 We will calculate: Rate of change of velocity with respect to time (or rate of change, for short). 4 Page 27

Finding a Formula for a Linear Function from a Table of Data t, time (secs) v, velocity (ft/sec) Page 27

Finding a Formula for a Linear Function from a Table of Data t, time (secs) v, velocity (ft/sec) ΔtΔt Page 27

Finding a Formula for a Linear Function from a Table of Data t, time (secs) v, velocity (ft/sec) ΔtΔtΔvΔv Page 27

Finding a Formula for a Linear Function from a Table of Data t, time (secs) v, velocity (ft/sec) ΔtΔtΔvΔvΔv/Δt Page 27

9

Since v is a function of t, we have: v = f(t). We also remember from Section 1.3: m = ? 10 Page 28

Since v is a function of t, we have: v = f(t). We also remember from Section 1.3: m = slope (or the rate of change = Δv/Δt) Here, m = ? 11 Page 28

Since v is a function of t, we have: v = f(t). We also remember from Section 1.3: m = slope (or the rate of change = Δv/Δt) Here, m = -32. So we have: v = b + mt or v = b -32t 12 Page 28

v = b -32t How do we solve for b? 13 Page 28

What can we use from this chart? t, time (secs) v, velocity (ft/sec) ΔtΔtΔvΔvΔv/Δt Page 28

t, time (secs)v, velocity (ft/sec) Take any pair of values from the chart: (1, 48) or (2, 16) or (3,-16) or (4,-48) and ? 15 Page 27

t, time (secs)v, velocity (ft/sec) Take any pair of values from the chart: (1, 48) or (2, 16) or (3,-16) or (4,-48) and substitute into: v = b -32t 16 Page 27

Take any pair of values from the chart: (1, 48) or (2, 16) or (3,-16) or (4,-48) and substitute into: v = b -32t (1,48): 48 = b - 32(1) → = b → b = Page 27

Take any pair of values from the chart: (1, 48) or (2, 16) or (3,-16) or (4,-32) and substitute into: v = b -32t (1,48): 48 = b - 32(1) → = b → b = 80 (2,16): 16 = b - 32(2) → = b → b = Page 27

Take any pair of values from the chart: (1, 48) or (2, 16) or (3,-16) or (4,-32) and substitute into: v = b -32t (1,48): 48 = b - 32(1) → = b → b = 80 (2,16): 16 = b - 32(2) → = b → b = 80 (3,-16): -16 = b - 32(3) → = b → b = Page 27

Take any pair of values from the chart: (1, 48) or (2, 16) or (3,-16) or (4,-32) and substitute into: v = b -32t (1,48): 48 = b - 32(1) → = b → b = 80 (2,16): 16 = b - 32(2) → = b → b = 80 (3,-16): -16 = b - 32(3) → = b → b = 80 (4,-48): -48 = b -32(4) → = b → b = Page 27

So what is our final equation? 21 Page 28

So what is our final equation? v = t 22 Page 28

v = t Note: m = -32 ft/sec per second (a.k.a. ft/sec 2 ) implies: the grapefruit’s velocity is decreasing by 32 ft/sec for every second that goes by. “The grapefruit is accelerating at -32 ft/sec per second.” Negative acceleration is also called deceleration. (Note: no shorthand way of saying “ft/sec per second”.) 23 Page 28

Finding a Formula for a Linear Function from a Graph We can calculate the slope, m, of a linear function using two points on its graph. Having found m, we can use either of the points to calculate b, the vertical intercept. 24 Page 28

Figure 1.25 shows oxygen consumption as a function of heart rate for two people.1.25 (a) Assuming linearity, find formulas for these two functions. (b) Interpret the slope of each graph in terms of oxygen consumption. 25 Page 28 Example 2

26 Page 28

Let's calculate m: 27 Page 29

Let's calculate m: 28 Page 29

Let's calculate m: 29 Page 29

What are our 2 linear equations so far? 30 Page 29

What are our 2 linear equations so far? For person A: 31 Page 29

What are our 2 linear equations so far? For person A: For person B: 32 Page 29

Now let's calculate b: 33 Page 29

Now let's calculate b: 34 Page 29

Now let's calculate b: 35 Page 29

What are our 2 linear equations? 36 Page 29

What are our 2 linear equations? For person A: 37 Page 29

What are our 2 linear equations? For person A: For person B: 38 Page 29

Figure 1.25 shows oxygen consumption as a function of heart rate for two people.1.25 (b) Interpret the slope of each graph in terms of oxygen consumption. What about (b)? 39 Page 29

Here are two reminders- this slide and the next: 40 Page 29

m=.01 m= Page 28

Since the slope for person B is smaller than for person A, person B consumes less additional oxygen than person A. 42 Page 29

We have $24 to spend on soda and chips for a party. A six-pack of soda costs $3 and a bag of chips costs $2. The number of six-packs we can afford, y, is a function of the number of bags of chips we decide to buy, x. (a) Find an equation relating x and y. (b) Graph the equation. Interpret the intercepts and the slope in the context of the party. 43 Page 30 Example 3

Let: x = # of bags of chips y = # of six-packs of soda 44 Page 30

Let: x = # of bags of chips $2x = amount spent on chips y = # of six-packs of soda 45 Page 30

Let: x = # of bags of chips $2x = amount spent on chips y = # of six-packs of soda $3y = amount spent on soda 46 Page 30

Let: x = # of bags of chips $2x = amount spent on chips y = # of six-packs of soda $3y = amount spent on soda & 2x + 3y = Page 30

2x + 3y = 24 Let's solve for y: 48 Page 30

49 Page 30

What is the slope and what is the y intercept? 50 Page 30

What is the slope and what is the y intercept? 51 Page 30

(b) Graph the equation. Interpret the intercepts and the slope in the context of the party. 52 Page 30

x (chips)y (soda) Let's plot some points: 53 Page N/A

x (chips)y (soda) Let's plot some points: 54 Page N/A

x (chips)y (soda) Let's plot some points: 55 Page N/A

56 Page 30

x (chips)y (soda) What conclusions can we draw from this table and from the equation below? 57 Page 31

x (chips)y (soda) What conclusions can we draw from this table and from the equation below? 58 Page 31

x (chips)y (soda) What conclusions can we draw from this table and from the equation below? 59 Page 31

x (chips)y (soda) What conclusions can we draw from this table and from the equation below? Page 31

x (chips)y (soda) What conclusions can we draw from this table and from the equation below? Page 31

Alternative Forms for the Equation of a Line: Slope-Intercept Form: y = b + mx [m=slope, y=y-int] Point-Slope Form: y – y o = m(x – x o ) [m=slope, (x o,y o ) point on the line] Standard Form: Ax + By + C = 0 [A, B, C constants] 62 Page 31

End of Section