Match the graphs with the functions. \( \begin{array}{ll}y=3 x^{2} & y=-|x|+4|y=|x-4| \quad y=|x|+4 \\ y=-4|x| & y=-|x+4| \quad y=-(x+3)^{2} \\ y=-4|x| & y=-x^{2}+3 \\ y=-3 x^{2} & y=(x-3)^{2}\end{array} \)

To match the graphs with the given functions, let's break down the functions based on their characteristics: 1. **Quadratic Functions** - \( y = 3x^2 \) is a classic upward-opening parabola with a vertex at the origin. - \( y = -x^2 + 3 \) is a downward-opening parabola with a vertex at (0, 3). - \( y = -(x+3)^2 \) is also a downward-opening parabola shifted left to -3. 2. **Absolute Value Functions** - \( y = -|x| + 4 \) is a V-shaped graph with the vertex at (0, 4) opening downwards. - \( y = |x| + 4 \) is a V-shaped graph starting at (0, 4) opening upwards. - \( y = -4|x| \) is a V-shaped graph opening downwards, steeper than \( y = -|x| + 4 \). 3. **Breaking down the remaining functions and considerations**: - \( y = |x - 4| \) is an absolute value function with its vertex at (4, 0). - \( y = (x - 3)^2 \) is an upward-opening parabola with its vertex at (3, 0). - \( y = -3x^2 \) is another downward-opening parabola, but it opens wider than \( -x^2 \). Now let's make the matches based on the descriptions: 1. **1st Graph**: \( y = 3x^2 \) - This is an upward parabola. 2. **2nd Graph**: \( y = -|x| + 4 \) - This is a downward opening V-shaped graph. 3. **3rd Graph**: \( y = |x| + 4 \) - This is an upward opening V-shaped graph starting at (0, 4). 4. **4th Graph**: \( y = -4|x| \) - This is a steeper downward V-shaped graph. 5. **5th Graph**: \( y = (x-3)^2 \) - This is an upward parabola with vertex at (3, 0). 6. **6th Graph**: \( y = -(x+3)^2 \) - This is a downward-opening parabola with vertex at (-3, 0). 7. **7th Graph**: \( y = -x^2 + 3 \) - This is another downward parabola vertex at (0, 3). 8. **8th Graph**: \( y = -3x^2 \) - Another downward parabola but opens wider than -x². By piecing together the characteristics, you should be able to establish which graphs correlate to which functions!