From the graph you provided, we can analyze the parabola and determine its equation. The graph shows a horizontally oriented parabola, where the vertex appears to be located at approximately $(1, 3)$, and the parabola opens to the right.

1. Vertex Form of the Parabola

The vertex form of a horizontally oriented parabola is given by: $y = a(x - h)^2 + k$ where:

• $(h, k)$ is the vertex of the parabola,
• $a$ determines the width and direction (opening left or right).

From the graph, it seems like the vertex is located at $(1, 3)$. So, the vertex form becomes: $y = a(x - 1)^2 + 3$ We now need to find the value of $a$. Looking at another point on the parabola, for instance $(5, 1)$ (as approximated from the graph), we can substitute this into the vertex form to solve for $a$.

Substitute $(x, y) = (5, 1)$ into the equation: $1 = a(5 - 1)^2 + 3$ $1 = a(4)^2 + 3$ $1 = 16a + 3$ $16a = 1 - 3 = -2$ $a = -\frac{2}{16} = -\frac{1}{8}$

Thus, the equation in vertex form is: $y = -\frac{1}{8}(x - 1)^2 + 3$

2. Standard Form of the Parabola

To convert the vertex form to the standard form $y = ax^2 + bx + c$, we expand the vertex form: $y = -\frac{1}{8}(x - 1)^2 + 3$ First, expand $(x - 1)^2$: $(x - 1)^2 = x^2 - 2x + 1$ Now substitute into the equation: $y = -\frac{1}{8}(x^2 - 2x + 1) + 3$ Distribute $-\frac{1}{8}$: $y = -\frac{1}{8}x^2 + \frac{1}{4}x - \frac{1}{8} + 3$ Simplify: $y = -\frac{1}{8}x^2 + \frac{1}{4}x + \frac{23}{8}$

Thus, the standard form is: $y = -\frac{1}{8}x^2 + \frac{1}{4}x + \frac{23}{8}$

Summary:

• Vertex form: $y = -\frac{1}{8}(x - 1)^2 + 3$
• Standard form: $y = -\frac{1}{8}x^2 + \frac{1}{4}x + \frac{23}{8}$

Do you need any further details or clarification?

Here are 5 related questions:

1. How do you determine the direction in which a parabola opens based on its equation?
2. What is the geometric significance of the coefficient $a$ in the equation of a parabola?
3. How do you derive the axis of symmetry for a parabola from its equation?
4. What is the relationship between the vertex form and the standard form of a quadratic equation?
5. How would the equation change if the parabola opened upwards instead of sideways?

Tip: Always check the vertex coordinates carefully when reading graphs to ensure accuracy when writing the equation.