1. **Stating the problem:** We have three sets of problems involving functions and points: - 9) Calculate function values for given $x$ values. - 10) Check if given points lie on the graphs of given functions. - 11) Find missing coordinates so points lie on the graphs. 2. **Formulas and rules:** - To find function values, substitute $x$ into the function formula. - To check if a point $(x,y)$ lies on a graph $y=f(x)$, verify if $y=f(x)$ holds true. - To find missing coordinates, use the function formula and solve for the unknown. --- ### Problem 9a) Calculate $y=3x^2$ for $x=\frac{1}{2}, -2, 0$: $$y=3x^2$$ - For $x=\frac{1}{2}$: $$y=3\left(\frac{1}{2}\right)^2=3\times\frac{1}{4}=\frac{3}{4}=0.75$$ - For $x=-2$: $$y=3(-2)^2=3\times4=12$$ - For $x=0$: $$y=3\times0^2=0$$ ### Problem 10a) Check if points $A(1,3)$ and $B(0,0)$ lie on $y=x^2$: - For $A(1,3)$: $$y=x^2=1^2=1 \neq 3$$ So $A$ does not lie on the graph. - For $B(0,0)$: $$y=0^2=0$$ So $B$ lies on the graph. ### Problem 11a) Find missing coordinates for $y=x^2$: - For $A(x,8)$: $$8=x^2 \Rightarrow x=\pm\sqrt{8}=\pm2\sqrt{2} \approx \pm2.828$$ - For $B(5,y)$: $$y=5^2=25$$ --- **Easiest way:** - Substitute known values directly into the function. - For checking points, plug in $x$ and compare with $y$. - For missing values, solve the equation for the unknown. This method applies similarly to all parts of problems 9, 10, and 11. Final answers for 9a, 10a, 11a: - 9a: $y(\frac{1}{2})=0.75$, $y(-2)=12$, $y(0)=0$ - 10a: $A$ no, $B$ yes - 11a: $x=\pm2.828$ for $A$, $y=25$ for $B$