For Problems \( 3-5 \), use tables on your calculator to help you investigate these quadratic functions. 3. Which of the following quadratics will have a maximum value at \( x=3 \) ? \( \begin{array}{ll}\text { (1) } y=x^{2}-6 x+19 & \text { (3) } y=-2 x^{2}+20 x-49 \\ \text { (2) } y=-4 x^{2}+24 x-21 & \text { (4) } y=2 x^{2}-3 x+7\end{array} \)

To find which of these quadratics has a maximum value at \( x = 3 \), we need to identify the parabolas that open downwards (as they can have a maximum). This occurs when the leading coefficient is negative. Let's analyze each function: 1. \( y = x^2 - 6x + 19 \) opens upwards (leading coefficient +1). 2. \( y = -4x^2 + 24x - 21 \) opens downwards (leading coefficient -4). 3. \( y = -2x^2 + 20x - 49 \) opens downwards (leading coefficient -2). 4. \( y = 2x^2 - 3x + 7 \) opens upwards (leading coefficient +2). Next, to find the vertex of the quadratics that open downwards, we use the vertex formula \( x = -\frac{b}{2a} \): - For (2): \( x = -\frac{24}{2 \cdot -4} = 3 \) (maximum at \( x = 3 \)) - For (3): \( x = -\frac{20}{2 \cdot -2} = 5 \) (not at \( x = 3 \)) Thus, the quadratic that has a maximum value at \( x = 3 \) is \( \text{(2) } y = -4x^2 + 24x - 21 \). --- To check the behavior around \( x = 3 \), you could use your calculator's table function to evaluate the values of these quadratics close to \( x = 3 \). Enter \( x = 2, 3, 4 \) and observe the output for each function. The quadratic \( y = -4x^2 + 24x - 21 \) will yield higher values at \( x = 3 \) compared to \( x = 2 \) and \( x = 4 \), confirming that it's indeed a maximum. Another useful approach is to analyze the symmetry of quadratic functions. The coefficient of \( x^2 \) reveals whether the parabola opens upwards or downwards, and when derived from the standard form, we can quickly determine \( x \) values where the vertex occurs. Familiarizing yourself with the properties of quadratics will certainly sharpen your problem-solving skills!