Page 17: Revision Class

let's solve the remaining questions from your document. I'll cover the "Revision Class" questions from page 17 and the "ASSIGNMENT" questions from page 18.

Page 17: Revision Class

1) Match each equation with its graph. Explain your choices.

• a) $y = x^2$ This is a standard parabola, symmetric about the y-axis. Among the given graphs, it is the least steep for $|x|>1$ and the steepest for $|x|<1$ compared to other even power functions. This matches graph h.
• b) $y = x^5$ This is an odd function, symmetric about the origin. It passes through $(0,0)$, $(1,1)$, and $(-1,-1)$. This matches graph g.
• c) $y = x^8$ This is an even function, symmetric about the y-axis. Compared to $y=x^2$, it is flatter near the origin (for $|x|<1$) and steeper for $|x|>1$. This matches graph f.

2) Classify each function.

• a) $f(x) = x^3 + 3x^2$ This is a sum of terms with non-negative integer powers of $x$. Classification: $\boxed{\text{Polynomial function (degree 3)}}$
• b) $g(t) = \cos^2 t - \sin t$ This function involves trigonometric ratios. Classification: $\boxed{\text{Trigonometric function}}$
• c) $r(t) = t\sqrt{3}$ This is a linear function, which is a type of polynomial. Classification: $\boxed{\text{Polynomial function (degree 1)}}$
• d) $V(t) = 8^t$ The variable $t$ is in the exponent. Classification: $\boxed{\text{Exponential function}}$
• e) $y = \frac{\sqrt{x}}{x^2+1}$ This function involves a root and a rational expression of polynomials. Classification: $\boxed{\text{Algebraic function}}$
• f) $g(u) = \log_{10} u$ This function involves a logarithm. Classification: $\boxed{\text{Logarithmic function}}$
• g) $f(t) = \frac{3t^2+2}{t}$ This is a ratio of two polynomials. Classification: $\boxed{\text{Rational function}}$
• h) $h(x) = 2 \cdot 3^x$ The variable $x$ is in the exponent. Classification: $\boxed{\text{Exponential function}}$
• i) $s(t) = \sqrt{t+4}$ This function involves a square root. Classification: $\boxed{\text{Root function}}$
• j) $y = x^4+5$ This is a sum of terms with non-negative integer powers of $x$. Classification: $\boxed{\text{Polynomial function (degree 4)}}$
• k) $g(x) = 3\sqrt[3]{x}$ This can be written as $3x^{1/3}$, which is a power function with a fractional exponent. Classification: $\boxed{\text{Power function (exponent } \frac{1}{3}\text{)}}$
• l) $y = \frac{1}{x^2}$ This can be written as $x^{-2}$, which is a power function with a negative integer exponent. Classification: $\boxed{\text{Power function (exponent -2)}}$

Page 18: ASSIGNMENT

1) On the same graph, plot the following. Discuss your observations.

• a) $y = x^2$

• b) $y = x^4$

• c) $y = x^6$

• Sketching:

  • All three graphs are parabolas opening upwards, symmetric about the y-axis.
  • All pass through the points $(0,0)$, $(1,1)$, and $(-1,1)$.
  • For $x \in (-1, 1)$, the graphs become flatter (closer to the x-axis) as the exponent increases. So, $y=x^6$ is below $y=x^4$, which is below $y=x^2$.
  • For $x \in (-\infty, -1) \cup (1, \infty)$, the graphs become steeper (further from the x-axis) as the exponent increases. So, $y=x^6$ is above $y=x^4$, which is above $y=x^2$.

• Observations:

  • All are even functions ($f(-x) = f(x)$), meaning their graphs are symmetric with respect to the y-axis.
  • All pass through $(0,0)$, $(1,1)$, and $(-1,1)$.
  • As the even exponent increases, the graph becomes flatter in the interval $(-1,1)$ and steeper outside this interval.

2) On the same $xy$-plane, draw the graph of. Discuss your observations.

• a) $y = x^3$

• b) $y = x^5$

• c) $y = x^7$

• Sketching:

  • All three graphs pass through the origin and have a similar S-shape.
  • All pass through the points $(0,0)$, $(1,1)$, and $(-1,-1)$.
  • For $x \in (-1, 1)$, the graphs become flatter (closer to the x-axis) as the exponent increases. So, $y=x^7$ is between $y=x^5$ and the x-axis, and $y=x^5$ is between $y=x^3$ and the x-axis.
  • For $x \in (-\infty, -1) \cup (1, \infty)$, the graphs become steeper (further from the x-axis) as the exponent increases. So, $y=x^7$ is further from the x-axis than $y=x^5$, which is further than $y=x^3$.

• Observations:

  • All are odd functions ($f(-x) = -f(x)$), meaning their graphs are symmetric with respect to the origin.
  • All pass through $(0,0)$, $(1,1)$, and $(-1,-1)$.
  • As the odd exponent increases, the graph becomes flatter in the interval $(-1,1)$ and steeper outside this interval.