怎么做---Advanced Math 2
Semester 2 Review
Chapter 5 Log/Exponents
Name: Wen jie Per. 7
1) Graph f(x) = (2)ˣ⁺² - 6
[Chart Description]:
Type: Cartesian Coordinate System Graph.
Axes: X-axis and Y-axis with numerical scales. X-axis scale appears to be from -6 to 6. Y-axis scale appears to be from -6 to 6.
Data Series: A curve is plotted, representing an exponential function. It approaches a horizontal asymptote below the x-axis (approximately y = -6) as x goes to negative infinity and increases rapidly as x goes to positive infinity. Several points are plotted on the curve, including what appears to be (-2, -5), (-1, -2), (0, -2), (1, 2), and (2, 10). There is an arrow indicating the direction of the curve.
2) Graph f(x) = log₂(x - 3) + 2
[Chart Description]:
Type: Cartesian Coordinate System Graph.
Axes: X-axis and Y-axis with numerical scales. X-axis scale appears to be from -6 to 6. Y-axis scale appears to be from -6 to 6.
Data Series: A curve is plotted, representing a logarithmic function. It approaches a vertical asymptote to the right of the y-axis (approximately x = 3) as x approaches 3 from the right and increases slowly as x goes to positive infinity. Several points are plotted on the curve, including what appears to be (4, 2), (5, 3), and (7, 4). There is an arrow indicating the direction of the curve.
3) What equation has the same end behavior as the graph below?
[Chart Description]:
Type: Cartesian Coordinate System Graph.
Axes: X-axis and Y-axis with numerical scales. X-axis scale appears to be from -6 to 6. Y-axis scale appears to be from -6 to 6.
Data Series: A curve is plotted, representing an exponential function. It approaches a horizontal asymptote at y = -4 as x goes to negative infinity and increases rapidly as x goes to positive infinity. Several points are plotted on the curve.
Options:
a) y = (0.5)ˣ - 4
b) y = (2)ˣ - 4
c) y = (0.5)⁻ˣ + 4
d) y = (2)⁻ˣ + 4
4) State whether the function models exponential growth or decay:
a) y = 200 (3/4)ˣ
b) y = 200 (4/3)ˣ
c) y = 200 (1.6)ˣ
d) y = 200 (0.6)ˣ
5) Rewrite in logarithmic form then solve for x.
64 = 2ˣ
Student work: x = 6
6) Rewrite in exponential form
log₃ x = 10
7) Solve for x, leave in terms of logs
4(3ˣ) = 16
8) Rewrite an equivalent expression using the change of base formula
log₃ 7 = x
9) Solve for x.
log₄ x = 5
10) What is the x-intercept of the graph?
f(x) = log(x)
11) The population of people in Illinois is changing according to the function p(t) = 12.5(0.87)ᵗ, where t is the time in years and p(t) is the number of people in millions. What can you state about how the population is changing?
The population is increasing/decreasing (choose one) by ___ % each year
Solve each equation for x.
12) log₅(2x - 9) = 0
Student work: 5⁰ = 2x - 9
x = 5
13) logₓ 4 = 1/2
Student work: x^(1/2) = 4
x = 16
14) log₄(3x + 4) - log₄ 12 = 0
Student work: log₄ (3x + 4) = log₄ 12
3x + 4 = 12 (rest of work not fully visible, but likely leads to 3x=8, x=8/3)
15) (1/3)²ˣ = 27ˣ⁻⁵
Student work: = (incomplete)
16) 2³ˣ⁻¹ = 12
Student work: log₂ 2³ˣ⁻¹ = log₂ 12
3x - 1 = log₂ 12
3x = log₂ 12 + 1
x = (log₂ 12 + 1) / 3
视频信息
答案文本
视频字幕
我们来学习指数函数的图像绘制。对于函数 f(x) = 2^(x+2) - 6，这是一个指数增长函数。底数 2 大于 1，所以函数递增。x+2 表示向左平移 2 个单位，减 6 表示向下平移 6 个单位。水平渐近线是 y = -6。我们计算几个关键点：当 x = -2 时，f(-2) = 2^0 - 6 = -5；当 x = 0 时，f(0) = 2^2 - 6 = -2；当 x = 1 时，f(1) = 2^3 - 6 = 2。
现在我们学习对数函数的图像。对于函数 f(x) = log₂(x-3) + 2，这是一个对数增长函数。x-3 表示向右平移 3 个单位，加 2 表示向上平移 2 个单位。垂直渐近线是 x = 3，定义域是 x > 3。我们计算几个关键点：当 x = 4 时，f(4) = log₂(1) + 2 = 2；当 x = 5 时，f(5) = log₂(2) + 2 = 3；当 x = 7 时，f(7) = log₂(4) + 2 = 4。
我们来分析终极行为。给定图像显示当 x 趋向负无穷时，y 趋向负 4；当 x 趋向正无穷时，y 趋向正无穷。我们逐一分析选项：选项 a 是 0.5 的 x 次方减 4，这是指数衰减，终极行为不符。选项 b 是 2 的 x 次方减 4，这是指数增长，当 x 趋向负无穷时 y 趋向负 4，当 x 趋向正无穷时 y 趋向正无穷，完全符合。选项 c 和 d 的终极行为也不符合。因此答案是选项 b。
我们来判断指数函数的增长性质。对于形式 y = a·b^x 的指数函数，关键看底数 b 的值。当 b 大于 1 时是指数增长，当 b 在 0 到 1 之间时是指数衰减。选项 a 中 b = 3/4 = 0.75 小于 1，是衰减。选项 b 中 b = 4/3 约等于 1.33 大于 1，是增长。选项 c 中 b = 1.6 大于 1，是增长。选项 d 中 b = 0.6 小于 1，是衰减。