Min Value Zeros 1C0C9A

1. **State the problem:** We are given the function $$f(x) = (x - 3)^2$$ and a graph of a parabola with vertex approximately at $$(-1, -2)$$ and x-intercepts approximately at $$(-2.4, 0)$$ and $$(0.4, 0)$$. We need to compare: - Which has the larger minimum value: the equation or the graph? - Which has more zeros: the equation or the graph? 2. **Analyze the equation's minimum value:** The function $$f(x) = (x - 3)^2$$ is a parabola opening upwards. The vertex form is already given, so the vertex is at $$x = 3$$. Calculate the minimum value: $$f(3) = (3 - 3)^2 = 0$$ So, the minimum value of the equation is $$0$$. 3. **Analyze the graph's minimum value:** The vertex of the graph is approximately at $$(-1, -2)$$. Since the parabola opens upward, the minimum value is the y-coordinate of the vertex: $$-2$$ 4. **Compare minimum values:** - Equation minimum value: $$0$$ - Graph minimum value: $$-2$$ Since $$-2 < 0$$, the graph has the smaller minimum value, so the equation has the larger minimum value. 5. **Analyze zeros of the equation:** Set $$f(x) = 0$$: $$ (x - 3)^2 = 0 $$ Solve: $$ x - 3 = 0 $$ $$ x = 3 $$ So, the equation has exactly one zero at $$x = 3$$. 6. **Analyze zeros of the graph:** The graph has x-intercepts approximately at $$-2.4$$ and $$0.4$$. So, the graph has two zeros. 7. **Compare zeros:** - Equation zeros: 1 - Graph zeros: 2 The graph has more zeros. **Final answers:** - The equation has the larger minimum value. - The graph has more zeros.