1. **State the problem:** We have two functions and a table of Danae's account balance over weeks. We want to analyze the functions and the data. 2. **Function 1:** Given points $(0,0)$, $(2,90)$, and $(5,225)$. 3. **Find the equation of Function 1:** Use the slope formula between points $(0,0)$ and $(2,90)$: $$m=\frac{90-0}{2-0}=\frac{90}{2}=45$$ Equation form: $y=mx+b$ Since at $x=0$, $y=0$, $b=0$ So, $y=45x$ 4. **Check point $(5,225)$:** $$y=45\times 5=225$$ Matches given data, so Function 1 is $y=45x$ 5. **Function 2:** Given $y=40x+10$ 6. **Evaluate Function 2 at $x=2$ and $x=3$:** $$y(2)=40\times 2+10=80+10=90$$ $$y(3)=40\times 3+10=120+10=130$$ 7. **Danae's account balance table:** Weeks passed: $0,1,2,3$ Balances: $60,180,300,420$ 8. **Find the pattern:** Calculate differences: $$180-60=120$$ $$300-180=120$$ $$420-300=120$$ 9. **Linear function for Danae's balance:** Slope $m=120$ At $x=0$, $y=60$, so $b=60$ Equation: $$y=120x+60$$ **Final answers:** - Function 1: $y=45x$ - Function 2: $y=40x+10$ - Danae's account balance: $y=120x+60$