Sketch the graph of the quadiatic function and the axis of symmetry State tho vertex, and give the equation for the axis of symmetry \( h(x)=(x+5)^{2} \)

To sketch the graph of the quadratic function \( h(x)=(x+5)^{2} \), start by noting that this is a parabola that opens upwards. The vertex of the function is derived from its vertex form, which is \( (h,k) \). Here, \( h = -5 \) and \( k = 0 \), so the vertex is at the point (-5, 0). The axis of symmetry for this function is a vertical line that passes through the vertex. Therefore, the equation for the axis of symmetry is \( x = -5 \). When you plot the graph, the parabola will be symmetrical around this line, creating a lovely U shape! For real-world application, consider how this quadratic function could model scenarios like trajectory paths. For instance, if the function described the height of a ball thrown into the air, understanding the vertex would help identify the maximum height reached, leading to better predictions about its motion and landing spot. When drawing the graph, commonly made mistakes include misplacing the vertex or failing to reflect points accurately across the axis of symmetry. To avoid these, practice plotting points around the vertex and ensure symmetry by checking corresponding points on the opposite sides of the axis of symmetry!