Question

question 5 · 1 point
determine the equation of the parabola whose graph is given below.
enter your answer in general form.

Explanation:

Step1: Identify vertex form

The vertex - form of a parabola is $y=a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. From the graph, the vertex $(h,k)=(3,4)$. So the equation is $y=a(x - 3)^2+4$.

Step2: Find the value of $a$

We can use another point on the parabola. Let's use the point $(4,3)$. Substitute $x = 4$ and $y = 3$ into $y=a(x - 3)^2+4$. We get $3=a(4 - 3)^2+4$, which simplifies to $3=a+4$. Solving for $a$, we subtract 4 from both sides: $a=3 - 4=-1$.

Step3: Convert to general form

Substitute $a=-1$ into $y=a(x - 3)^2+4$. First, expand $(x - 3)^2=x^{2}-6x + 9$. Then $y=-(x^{2}-6x + 9)+4$. Distribute the - 1: $y=-x^{2}+6x-9 + 4$. Combine like - terms: $y=-x^{2}+6x-5$. In general form $Ax^{2}+Bx + C=0$, we move all terms to one side to get $x^{2}-6x + y+5 = 0$.

Answer:

$x^{2}-6x + y+5 = 0$