\( h(r)=(r+1)(r+8) \) 1) What are the zeros of the function? Write the smaller \( r \) first, and the larger \( r \) second. smaller \( r= \) larger \( r= \) 2) What is the vertex of the parabola? (

To find the zeros of the function \( h(r)=(r+1)(r+8) \), we set it equal to zero: \((r+1)(r+8) = 0\) This gives us two solutions: \( r + 1 = 0 \) leading to \( r = -1 \) and \( r + 8 = 0 \) giving \( r = -8 \). So, the smaller \( r = -8 \) and the larger \( r = -1 \). 1) smaller \( r= -8 \) larger \( r= -1 \) Now, for the vertex of the parabola, we can rewrite the function in standard form. The vertex \( r_v \) of a parabola defined as \( ax^2 + bx + c \) can be found using the formula \( r_v = -\frac{b}{2a} \). Expanding \( h(r) \) gives us \( r^2 + 9r + 8 \). Here, \( a=1 \) and \( b=9 \). So, \( r_v = -\frac{9}{2(1)} = -\frac{9}{2} \). To find the corresponding \( h(-\frac{9}{2}) \): \[ h\left(-\frac{9}{2}\right) = \left(-\frac{9}{2}+1\right)\left(-\frac{9}{2}+8\right) = \left(-\frac{7}{2}\right)\left(\frac{7}{2}\right) = -\frac{49}{4}. \] Thus, the vertex is: 2) Vertex: \( \left( -\frac{9}{2}, -\frac{49}{4} \right) \) Now you're ready to wow everyone with your knowledge of parabolas! As a fun fact, did you know that the symmetry of the parabola allows you to reflect points across its vertex? This property makes graphing and understanding their shape much more manageable! Plus, parabolas appear in nature, from the trajectory of a thrown ball to the shape of satellite dishes, showing just how versatile this mathematical concept is!