Certainly! To find these points, we will need to complete several steps, including finding the roots of the equation, determining the vertex of the parabola, and choosing additional two points for plotting.
Firstly, we need to find the roots of the equation y = x² + 10x + 24. By solving this quadratic equation, we obtain two roots -6.0 and -4.0. These are the x-values where the function intersects the x-axis, resulting in y equals zero, thus our two roots create the points (-6.0, 0) and (-4.0, 0).
Next, we have to locate the vertex of the parabola. The vertex is the point on the graph where the function reaches its maximum or minimum value. In this case, since the coefficient of x² is positive, the parabola opens upwards, meaning the vertex will be a minimum point. The x-coordinate of the vertex can be found by using the formula -b/2a, where a is the coefficient of x² and b is the coefficient of x. By substituting a=1 and b=10 into the formula, the x-coordinate of the vertex is found to be -5.0. To find the y-coordinate, substitute -5.0 into the equation for x, and you'll get the vertex value -1.0. So, the vertex in this case is the point (-5.0, -1.0).
Finally, to get more points for graphing, I pick two additional points close to the vertex, specifically, one unit to the left and one unit to the right of the x-value of the vertex. Therefore, the x-values for these points are -6.0 and -4.0 respectively. We use these x-values and plug them into the equation for x to get the associated y-values, and in doing so, we find our points to be (-6.0, 0.0) and (-4.0, 0.0), both of which coincide with our already-found roots of the equation.
In conclusion, the five points that are consequential for the plotting of the function y = x² + 10x + 24 are (-6.0, 0), (-4.0, 0), (-5.0, -1.0), (-6.0, 0.0), and (-4.0, 0.0).