Question

A quadratic function has one root at 5 and has a vertex point at (7, 2).

Answer 1

Explanation

• Vertex Form

$y = a {(x - h)}^{2} + k$

where a-term determines the shape of graph.

h-term determines the change of graph for x-axis.

k-term determines the change of graph for y-axis.

Vertex of the graph is at (h,k).

• Substitute the vertex value in the equation.

$y = a {(x - 7)}^{2} + 2$

We need to find the value of a-term. We have the given root which we can substitute in the equation.

Also the roots are on x-axis, meaning that the y-value for roots must be 0.

• Substitute (5,0) in the equation.

$0 =a {(5 - 7)}^{2} + 2 \\ 0 =a {( - 2)} ^{2} + 2 \\ 0 = 4a + 2$

• Solve for a-term.

$4a + 2 = 0 \\ 4a = - 2 \\ a = \frac{ - 2}{4} \\ a = - \frac{1}{2}$

Therefore the value of a is - 1/2. Rewrite the equation as we get the answer.

Answer

$\large \boxed{y = - \frac{1}{2} {(x - 7)}^{2} + 2}$