1. **Problem:** Match each graph with the correct equation from the given list. 2. **Step 1: Identify the type of graph and its features.** - Parabolas open upwards or downwards and are quadratic functions. - Lines are linear functions. - Inequalities are represented by shaded regions above or below lines. 3. **Step 2: Analyze each graph description and match with equations:** - Top-left graph: Parabola opening upwards with vertex between $x=1$ and $x=2$. - Candidates: (a) $y=5x^2 - 8x + 4$ or (b) $y=5x^2 - 8x - 4$ - Top-right graph: Linear function with shaded region below a line with negative slope. - Candidates: (k) $y < -5x - 4$ or (l) $y \leq -5x - 4$ - Middle-left graph: Straight line with negative slope crossing both axes. - Candidates: (g) $y = -5x + 4$ or (h) $y = -5x - 4$ - Middle-right graph: Downward opening parabola with vertex around $x=1$. - Candidates: (c) $y = -5x^2 - 8x + 4$ or (d) $y = -5x^2 - 8x - 4$ - Bottom-left graph: Linear function with shaded region above a line with positive slope. - Candidates: (m) $y > 5x + 4$, (n) $y \geq 5x + 4$, (o) $y > 5x - 4$, or (p) $y \leq 5x - 4$ - Bottom-right graph: Straight line with positive slope crossing the origin. - Candidates: (e) $y = 5x + 4$ or (f) $y = 5x - 4$ 4. **Step 3: Use vertex and intercept information to select the exact equations:** - For parabolas, vertex formula is $x = -\frac{b}{2a}$. - Top-left parabola: $a=5$, $b=-8$, so vertex at $x = -\frac{-8}{2\times5} = \frac{8}{10} = 0.8$ (close to 1), so (a) or (b). - Check $y$-intercept: For $x=0$, (a) $y=4$, (b) $y=-4$. - Since vertex is between 1 and 2 and parabola opens upwards, and $y$-intercept is positive, choose (a). - Top-right graph: shaded below line with negative slope and intercept -4. - Inequality $y \leq -5x - 4$ matches (l). - Middle-left graph: line with negative slope crossing both axes. - Check intercepts for (g) and (h): - (g) $y = -5x + 4$ intercepts at $y=4$ and $x=\frac{4}{5}=0.8$. - (h) $y = -5x - 4$ intercepts at $y=-4$ and $x=-\frac{4}{5}=-0.8$. - Since it crosses both axes positively, choose (g). - Middle-right parabola: downward opening, $a=-5$, $b=-8$. - Vertex at $x = -\frac{-8}{2\times -5} = -\frac{-8}{-10} = -0.8$. - Check $y$-intercept: (c) $y=4$, (d) $y=-4$. - Vertex around $x=1$ and $y$-intercept positive, choose (c). - Bottom-left graph: shaded above line with positive slope. - Inequality $y \geq 5x - 4$ matches (p). - Bottom-right graph: line with positive slope crossing origin. - Check intercepts for (e) and (f): - (e) $y=5x+4$ intercepts at $y=4$. - (f) $y=5x-4$ intercepts at $y=-4$. - Since it crosses origin, intercept is 0, so neither exactly matches, but closest is (f) with intercept -4. 5. **Final answers:** - Top-left graph: (a) $y=5x^2 - 8x + 4$ - Top-right graph: (l) $y \leq -5x - 4$ - Middle-left graph: (g) $y = -5x + 4$ - Middle-right graph: (c) $y = -5x^2 - 8x + 4$ - Bottom-left graph: (p) $y \geq 5x - 4$ - Bottom-right graph: (f) $y = 5x - 4$