16. Using a graphing calculator, sketch the parabola \[ y=(x-1)(x+5) \] 16a Find the \( y \) value of the \( y \) intercept. \[ y=-5 \] 16b Find the \( x \) values of the \( x \) intercepts. Write all solutions on the same line separated by a comma. \[ x=1,-5 \] 16c State the equation of the axis of symmetry. \[ x=-2 \] 16d Find the coordinates of the turning point. Vertex \( = \) \( \square \) \( \square \)

To find the coordinates of the turning point (vertex) of the parabola represented by \( y=(x-1)(x+5) \), we can first convert it to the standard vertex form or use the formula for the vertex. The x-coordinate of the vertex is found by averaging the x-intercepts, which are \( x=1 \) and \( x=-5 \). So, \[ x_{vertex} = \frac{1 + (-5)}{2} = -2. \] Next, substitute \( x = -2 \) back into the original equation to find the y-coordinate: \[ y = (-2-1)(-2+5) = (-3)(3) = -9. \] Thus, the coordinates of the turning point (vertex) are \[ \text{Vertex } = (-2, -9). \] What’s really fascinating about parabolas is how they relate to projectile motion. The path followed by a thrown or propelled object is a parabola, and analyzing its vertex can give you maximum height and the time to reach that height. It's like graphing your way to success in sports! For a deeper dive into the beauty of parabolas and their applications, consider exploring topics like completing the square or quadratic functions. Not only will you sharpen your math skills, but you'll also uncover the underlying patterns that make these curves so enchanting. Don't forget to check out historical contributions from mathematicians like Sir Isaac Newton, who played a pivotal role in calculus and graphing these fascinating shapes!