1.  [latex]h(y) = 2y⁴ - 3y² + 5y - 7[/latex]
   • Degree: [latex]4[/latex]
   • Leading Term: [latex]2y⁴[/latex]
   • Leading Coefficient: [latex]2[/latex]
2. [latex]k(s) = -0.5s³ + 2s - 1[/latex]
   • Degree: [latex]3[/latex]
   • Leading Term: [latex]-0.5s³[/latex]
   • Leading Coefficient: [latex]-0.5[/latex]
3. [latex]m(w) = w⁵ + 3w⁴ - 2w² + w[/latex]
   • Degree: [latex]5[/latex]
   • Leading Term: [latex]w⁵[/latex]
   • Leading Coefficient: [latex]1[/latex] (remember, when not written, the coefficient is [latex]1[/latex])

Set the function equal to zero:

[latex]2x² + 5x - 3 = 0[/latex]

We’ll use the quadratic formula since this doesn’t factor easily. Here [latex]a = 2, b = 5[/latex], and [latex]c = -3[/latex]

Apply the quadratic formula:

[latex]\begin{align*}
x &= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\
x &= \frac{-5 \pm \sqrt{5^2 - 4(2)(-3)}}{2(2)} \\
x &= \frac{-5 \pm \sqrt{25 + 24}}{4} \\
x &= \frac{-5 \pm \sqrt{49}}{4} \\
x &= \frac{-5 \pm 7}{4}
\end{align*}[/latex]

Simplify:

[latex]\begin{align*} x &= \frac{(-5 + 7)}{4} \text{ or } \frac{(-5 - 7)}{4} \\[6pt] x &= \frac{2}{4} \text{ or } \frac{-12}{4} \\[6pt] x &= \frac{1}{2} \text{ or } -3 \end{align*}[/latex]

We can check our solutions by plugging them back in for [latex]x[/latex] and solving to see if the answer is [latex]0[/latex]:

[latex]g(\frac{1}{2}) = 2(\frac{1}{2})² + 5(\frac{1}{2}) - 3 = \frac{1}{2} + \frac{5}{2} - 3 = 0[/latex]

[latex]g(-3) = 2(-3)² + 5(-3) - 3 = 18 - 15 - 3 = 0[/latex]

Therefore, the zeros of [latex]g(x)[/latex] are [latex]x = \frac{1}{2}[/latex] and [latex]x = -3[/latex].

1. Identify key features:
   • Degree: [latex]5[/latex] (odd)
   • Leading coefficient: [latex]-2[/latex] (negative)
2. Describe end behavior:
   • As [latex]x[/latex] approaches positive infinity, [latex]f(x)[/latex] approaches negative infinity
   • As [latex]x[/latex] approaches negative infinity, [latex]f(x)[/latex] approaches negative infinity
3. Basic shape:
   • The graph will have one arm pointing down on the right side
   • The other arm will point up on the left side
   • There may be turns or oscillations between these extremes

Graph one linear function for [latex]x \le 2[/latex] and then graph a different linear function for [latex]x>2[/latex].

The piecewise-defined function is constant on the intervals [latex](0,1], \, (1,2], \, \cdots[/latex]

[latex]C(x)=\begin{cases} 49, & 0 < x \le 1 \\ 70, & 1 < x \le 2 \\ 91, & 2 < x \le 3 \end{cases}[/latex]

Algebraic

1.  Algebraic; It’s a ratio of polynomials.
2. Transcendental;  Involves sine and cosine, which are transcendental.
3. Mixed – Both algebraic and transcendental; [latex]\sqrt[3]{(x² + 1)}[/latex] is algebraic (root function), but [latex]log₃(x)[/latex] is transcendental.
4. Transcendental;  [latex]2^x[/latex] and [latex]3^x[/latex] are exponential (transcendental), and the overall function can’t be simplified to an algebraic form.

• Identify the base function: [latex]f(x) = \sqrt{x}[/latex]
• Break down the transformations:
  • Inside the square root:[latex](x - 1)[/latex] indicates a horizontal shift
  • Outside the square root: [latex]2[/latex] indicates vertical scaling
  • [latex]+3[/latex] at the end indicates a vertical shift
• Apply transformations in order:
  • Horizontal shift: [latex]\sqrt{(x - 1)}[/latex] shifts the graph [latex]1[/latex] unit right
  • Vertical scaling: [latex]2\sqrt{(x - 1)}[/latex] stretches the graph vertically by a factor of [latex]2[/latex]
  • Vertical shift: [latex]2\sqrt{(x - 1)} + 3[/latex] shifts the graph up [latex]3[/latex] units
• Describe the final graph:
  • The graph of [latex]g(x)[/latex] is the square root function shifted [latex]1[/latex] unit right, stretched vertically by a factor of [latex]2[/latex], and shifted up [latex]3[/latex] units.
  • The domain is [latex][1, ∞)[/latex] because [latex]x - 1[/latex] must be non-negative under the square root.
  • The range is [latex][3, ∞)[/latex] due to the vertical stretch and shift.

We can check this by plugging both equations into a graphing software.