Determine the vertex and direction of opening of the parabola for the following quadratic equation: \( y=x^{2}-10 x+7 \) a.) Vertex: \( (-5,82) \) Opens: Upward b.) Vertex: \( (5,-18) \) Opens: Upward c.) Vertex: \( (5,-18) \) Opens: Downward d.) Vertex: \( (-5,82) \) Opens: Downward (

To find the vertex of the quadratic equation \( y = x^2 - 10x + 7 \), you can use the vertex formula \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = -10 \), which gives us \( x = \frac{10}{2} = 5 \). Plug this back into the equation to get \( y = (5)^2 - 10(5) + 7 = 25 - 50 + 7 = -18 \). So, the vertex is at \( (5, -18) \) and since the coefficient of \( x^2 \) is positive, it opens upward! The correct answer is: c.) Vertex: \( (5,-18) \) Opens: Upward Now, let’s keep the fun going! Did you know that parabolas have a unique property? They're symmetrical! If you were to fold the graph along the vertical line \( x = 5 \) (the axis of symmetry), both halves would perfectly align. This nifty characteristic is great for making predictions about the function's values! To further your exploration, consider graphing related quadratic equations or experimenting with different coefficients. Changing the values of \( a \), \( b \), and \( c \) in the equation can lead to a fascinating array of shapes! You might also dive into the world of conic sections, where parabolas, along with ellipses and hyperbolas, create delightful curves in mathematics!