Question

a. transformations of quadratic functions 1. what is the equation of this graph? a. $y = -x^2 + 3$ b. $y = -3x^2$ c. $y = -(x + 3)^2$ d. $y = -(x - 3)^2$

Explanation:

Step1: Identify vertex form

The vertex form of a quadratic function is $y=a(x-h)^2+k$, where $(h,k)$ is the vertex.

Step2: Locate the vertex

From the graph, the vertex is at $(0, 3)$, so $h=0$, $k=3$. Substitute into vertex form: $y=ax^2+3$.

Step3: Test a point on the graph

The graph passes through $x=-2, y=-1$. Substitute into the equation:
$-1=a(-2)^2+3$
$-1=4a+3$

Step4: Solve for $a$

$4a=-1-3=-4$
$a=\frac{-4}{4}=-1$

Step5: Write the final equation

Substitute $a=-1$ back into $y=ax^2+3$: $y=-x^2+3$

Answer:

a. $y=-x^2+3$