HallsofIvy said:Why? If you are posting this because you want help with it then you should show us what you do understand about it yourself so that we will know what kinds of hints and help you need.
Do you understand what a "normal" to a graph is? Do you understand that a normal to a graph, at a point, is perpendicular to the tangent to that graph at that point? Can you find the tangent to $y= 2x^2+ 4x+ \frac{7}{4}$.
Notice that $2(3)^2+ 4(3)+ \frac{7}{2}= 18+ 12+ \frac{7}{4}= 30+ \frac{7}{4}= \frac{127}{4}$ not $\frac{15}{2}$ so the given point is not on the curve. You will need to find the tangent line at some point $\left(a, 2a^2+ 4a+ \frac{7}{2}\right)$ then find a so that tangent line passes through (3, 15/4). Once you have found the point and the equation of the tangent, it should be easy to find the normal line.
The normal of a parabola is a line that is perpendicular to the tangent line at a given point on the parabola.
The normal of a parabola can be calculated by finding the slope of the tangent line at a given point and then taking the negative reciprocal of that slope.
The normal of a parabola is important because it allows us to find the equation of the tangent line at a given point on the parabola, which can be useful in solving real-world problems involving parabolic curves.
No, a parabola can only have one normal at a given point. This is because the normal is always perpendicular to the tangent line, and there can only be one line that is perpendicular to a given line at a specific point.
To find the equation of the normal to a parabola, you first need to find the slope of the tangent line at a given point on the parabola. Then, you can use the point-slope form of a line to write the equation of the normal.