The graph of a quadratic function is given. Write the function's equation, selecting from the following options.
$f\left(x\right)=x^2+2x+1g\left(x\right)=x^2-2x+1$
$h\left(x\right)=x^2-1j\left(x\right)=-x^2-1$

Verified step by step guidance

1. Step 1: Identify the general form of the quadratic function, which is \(y = ax^2 + bx + c\).
2. Step 2: Use the point where the graph crosses the y-axis, which is the y-intercept. From the graph, this point is \((0, -20)\), so substitute \(x=0\) and \(y=-20\) into the equation to find \(c\). This gives \(-20 = a(0)^2 + b(0) + c\), so \(c = -20\).
3. Step 3: Use the other given point on the graph, \((3, -2)\), and substitute \(x=3\), \(y=-2\), and \(c=-20\) into the equation: \(-2 = a(3)^2 + b(3) - 20\).
4. Step 4: Simplify the equation from Step 3 to get \(-2 = 9a + 3b - 20\). Rearrange it to \$9a + 3b = 18$.
5. Step 5: To find \(a\) and \(b\), you need another equation. Use the vertex form or the fact that the vertex is at \(x=2\) (since the maximum point is between 2 and 3 on the graph). Use the vertex formula \(x = -\frac{b}{2a}\) to set up the second equation: \(2 = -\frac{b}{2a}\). Solve this system of two equations to find \(a\) and \(b\).

Key Concepts

Quadratic Function and Its Standard Form

A quadratic function is a polynomial of degree two, typically written as f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola that opens upward if a > 0 and downward if a < 0. Understanding the standard form helps in identifying the coefficients and how they affect the shape and position of the parabola.

Using Points to Determine the Quadratic Equation

Given points on the graph, such as (0, -20) and (3, -2), you can substitute these coordinates into the quadratic equation to form a system of equations. This system can be solved to find the values of a, b, and c, which define the specific quadratic function that fits the graph.

Interpreting the Y-Intercept and Vertex

The y-intercept of a quadratic function is the point where the graph crosses the y-axis, given by (0, c). The vertex is the highest or lowest point on the parabola, indicating the maximum or minimum value of the function. Identifying these points from the graph helps in writing or verifying the quadratic equation.