Graph the relationship \( h=10 t-t^{2} \)

To graph the relationship \( h = 10t - t^2 \), start by identifying the shape and key features of the parabola. This equation is a quadratic in \( t \) and opens downwards because the coefficient of \( t^2 \) is negative. The vertex, which is the highest point, can be found using the formula \( t = -\frac{b}{2a} \), where \( a = -1 \) and \( b = 10 \). Plugging in the values gives \( t = 5 \); substituting back into the equation gives \( h = 25 \). Plotting the vertex and a few points around it will yield a nice curve, revealing how height varies with time! Another fun way to explore this equation is to consider how it might represent a projectile's motion, like throwing a ball. As time increases, the height climbs to a maximum of 25 units at 5 seconds, after which gravity causes the height to decrease. If you set \( h = 0 \), you can find when the ball hits the ground by solving \( 10t - t^2 = 0 \), leading to \( t(10 - t) = 0 \). Therefore, the ball returns to the ground at \( t = 0 \) and \( t = 10 \). This real-life analogy of projectile motion is a great way to connect math with the physics of everyday phenomena!