To solve the system of equations by graphing, follow these steps:

Step 1: Identify the Equations

We are given:

1. $y = \frac{1}{2}(x - 3)^2 + 1$ → This is a parabola that opens upward with its vertex at $(3,1)$.
2. $y = -x + 8$ → This is a straight line with a slope of -1 and a y-intercept at $(0,8)$.

Step 2: Graph Each Equation

1. Parabola $y = \frac{1}{2}(x - 3)^2 + 1$

   • Vertex: $(3,1)$
   • Direction: Opens upward because the coefficient $\frac{1}{2}$ is positive.
   • Additional Points:
     • At $x = 1$, $y = \frac{1}{2}(1-3)^2 +1 = \frac{1}{2}(4) +1 = 3$ → Point $(1,3)$
     • At $x = 5$, $y = \frac{1}{2}(5-3)^2 +1 = \frac{1}{2}(4) +1 = 3$ → Point $(5,3)$

2. Line $y = -x + 8$

   • Y-intercept: $(0,8)$
   • Slope: -1 (moves down 1 unit for every 1 unit to the right)
   • Additional Points:
     • At $x = 1$, $y = -1 + 8 = 7$ → Point $(1,7)$
     • At $x = 2$, $y = -2 + 8 = 6$ → Point $(2,6)$

Step 3: Find Intersection Points

By solving algebraically, set both equations equal to each other:

$\frac{1}{2}(x - 3)^2 + 1 = -x + 8$

1. Subtract 1 from both sides:

$\frac{1}{2}(x - 3)^2 = -x + 7$

2. Multiply everything by 2 to eliminate the fraction:

$(x - 3)^2 = -2x + 14$

3. Expand:

$x^2 - 6x + 9 = -2x + 14$

4. Move everything to one side:

$x^2 - 4x - 5 = 0$

5. Factor:

$(x - 5)(x + 1) = 0$

6. Solve for $x$:

$x = 5 \quad \text{or} \quad x = -1$

7. Find corresponding $y$-values using $y = -x + 8$:

   • If $x = 5$, then $y = -5 + 8 = 3$ → Point $(5,3)$.
   • If $x = -1$, then $y = -(-1) + 8 = 9$ → Point $(-1,9)$.

Final Answer:

The system of equations intersects at (5,3) and (-1,9).

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Would you like a graph of the solution? Also, do you need a step-by-step explanation of any part?

Related Questions:

1. How do you determine the vertex of a parabola?
2. What are the different ways to solve a system of equations?
3. How do you factor quadratic equations?
4. What are the key properties of a parabola?
5. How do you determine the slope and y-intercept of a linear equation?

Tip:

When solving systems of equations graphically, always check for symmetry in parabolas and use a table of values to ensure accurate plotting.