1. **State the problem:** We need to find the equations for four graphs based on their shapes and points. 2. **Top-left graph:** It is a cubic curve passing through (-2, -8), flattening near (0,0), and rising steeply at (2,8). - The cubic function formula is $y = x^3$. - Check points: $(-2)^3 = -8$, $(0)^3 = 0$, $(2)^3 = 8$. - This matches the graph perfectly. 3. **Top-right graph:** It is a straight line passing through (-2, 2) and (2, -2). - The linear function formula is $y = mx + b$. - Calculate slope $m = \frac{-2 - 2}{2 - (-2)} = \frac{-4}{4} = -1$. - Since it passes through (0,0), $b=0$. - Equation is $y = -x$. 4. **Bottom-left graph:** It is a downward parabola with vertex at (0,4) passing through (-2,0) and (2,0). - The quadratic function formula is $y = ax^2 + bx + c$. - Vertex form: $y = a(x - h)^2 + k$ with vertex $(h,k) = (0,4)$, so $y = a x^2 + 4$. - Use point (-2,0): $0 = a(-2)^2 + 4 \Rightarrow 0 = 4a + 4 \Rightarrow 4a = -4 \Rightarrow a = -1$. - Equation is $y = -x^2 + 4$. 5. **Bottom-right graph:** It is a hyperbola with horizontal asymptote $y = -1$, vertical asymptote $x=0$, passing through (2,0.5) and (-2,-0.5). - The function form is $y = \frac{k}{x} + c$. - Horizontal asymptote $y = c = -1$. - Use point (2,0.5): $0.5 = \frac{k}{2} - 1 \Rightarrow \frac{k}{2} = 1.5 \Rightarrow k = 3$. - Equation is $y = \frac{3}{x} - 1$. **Final answers:** - Top-left: $y = x^3$ - Top-right: $y = -x$ - Bottom-left: $y = -x^2 + 4$ - Bottom-right: $y = \frac{3}{x} - 1$