To find the standard form of the quadratic function $y = ax^2 + bx + c$ given the graph, we need to determine the values of $a$, $b$, and $c$.

Observations:

1. The value of $c$:

   • The $y$-intercept (where $x = 0$) is given as 3, so $c = 3$.

2. The value of $a$:

   • Since the parabola opens downwards, $a$ is negative.
   • We are given $a = -1$ in the answer space, so we assume $a = -1$.

3. Finding $b$:

   • To find $b$, we can use other points on the graph along with $a = -1$ and $c = 3$.

From the graph, it appears the vertex is located at $(-3, 4)$.

Vertex Form Conversion:

The vertex form of a quadratic function is: $y = a(x - h)^2 + k$ where $(h, k)$ is the vertex. Substituting $a = -1$, $h = -3$, and $k = 4$: $y = -1(x + 3)^2 + 4$

Expanding this to get the standard form: $y = -1(x^2 + 6x + 9) + 4$ $y = -x^2 - 6x - 9 + 4$ $y = -x^2 - 6x - 5$

Final Answer:

The standard form of the quadratic function is: $y = -x^2 - 6x - 5$

Thus:

• $a = -1$
• $b = -6$
• $c = 3$

Would you like further details on this solution?

Related Questions:

1. How do you determine the vertex of a parabola?
2. How can you tell if a parabola opens upwards or downwards?
3. What are the steps to convert from vertex form to standard form?
4. How does the value of $a$ affect the width of a parabola?
5. How do you find the $y$-intercept from a graph?

Tip:

To quickly find $b$ in standard form, identify the vertex and substitute points into the equation.