1. In the first line, type the (implicit) function: \( x+y=3 \) 2. In the second line, type the (explicit) function \( (t)=-*)+5 \) 3. Celcutate (by hand) the intercepts for \( x+y=3 \). \[ \begin{array}{cc} x+0=3 & o+y=3 \\ x=3 & y=3 \end{array} \] The \( y \)-intercept(s) of \( x+y=3 \) is/are: \( (0,3) \) The \( x \)-intercept(s) of \( x+y=3 \) is/are: \( (3,0) \) Once you find the intercept(s) algebraically, confirm your intercepts using Desmos. 4. Calculate (by hand) the intercepts for \( f(x)=-x^{2}+5 \). The \( y \)-intercept(s) of \( f(x) \) is/are: The \( x \)-intercept(s) of \( f(x) \) is/are: Once you find the intercept(s) algebraically, confirm your intercepts using Desmos. 5. Calculate (by hand) the point(s) of intersection of the two graphs.

Below is a step‐by‐step solution for the problem. ────────────────────────────── Step 1. Write the Implicit Function Write the equation in its given form:   x + y = 3 ────────────────────────────── Step 2. Write the Explicit Function The explicit function is given by:   f(x) = –x² + 5 ────────────────────────────── Step 3. Find the Intercepts for x + y = 3 • To find the y‐intercept, set x = 0:   0 + y = 3 ⟹ y = 3   The y‐intercept is: (0, 3) • To find the x‐intercept, set y = 0:   x + 0 = 3 ⟹ x = 3   The x‐intercept is: (3, 0) Confirm these intercepts using Desmos by graphing the equation x + y = 3. ────────────────────────────── Step 4. Find the Intercepts for f(x) = –x² + 5 • To find the y‐intercept, substitute x = 0:   f(0) = –(0)² + 5 = 5   The y‐intercept is: (0, 5) • To find the x‐intercepts, set f(x) = 0 and solve:   –x² + 5 = 0 ⟹ x² = 5 ⟹ x = ±√5   The x‐intercepts are: (√5, 0) and (–√5, 0) Again, you can confirm these intercepts on Desmos by graphing f(x) = –x² + 5. ────────────────────────────── Step 5. Find the Point(s) of Intersection of the Two Graphs The two equations are:   (1) x + y = 3   (2) y = –x² + 5 Use substitution. From (1), express y in terms of x:   y = 3 – x Replace in (2):   3 – x = –x² + 5 Solve for x:   –x² + 5 = 3 – x   Bring all terms to one side:   –x² + 5 – 3 + x = 0   –x² + x + 2 = 0 Multiply the entire equation by –1 to simplify:   x² – x – 2 = 0 Factor the quadratic:   (x – 2)(x + 1) = 0 Thus, the solutions for x are:   x = 2  or  x = –1 Find the corresponding y-values using y = 3 – x: • For x = 2:   y = 3 – 2 = 1   Intersection point: (2, 1) • For x = –1:   y = 3 – (–1) = 4   Intersection point: (–1, 4) Confirm these intersection points on Desmos by graphing both equations. ────────────────────────────── Summary of Answers 1. Implicit function:   x + y = 3 2. Explicit function:   f(x) = –x² + 5 3. Intercepts for x + y = 3:   y‐intercept: (0, 3)   x‐intercept: (3, 0) 4. Intercepts for f(x) = –x² + 5:   y‐intercept: (0, 5)   x‐intercepts: (√5, 0) and (–√5, 0) 5. Intersection points of the two graphs:   (2, 1) and (–1, 4) Use Desmos to verify these results by graphing the equations. Feel free to ask if you have any questions or need further explanation!