The equation given in the image is $f(x) = -x^2 + 4x + 3$, which is a quadratic function. To identify the vertex and the y-intercept, let's break it down.

1. Finding the Vertex:

The equation is in standard form $f(x) = ax^2 + bx + c$. For a quadratic function $ax^2 + bx + c$, the vertex $x$-coordinate is given by: $x = \frac{-b}{2a}$ In your equation, $a = -1$, $b = 4$, and $c = 3$. So: $x = \frac{-4}{2(-1)} = 2$ Now, substitute $x = 2$ back into the equation to find the $y$-coordinate of the vertex: $f(2) = -(2)^2 + 4(2) + 3 = -4 + 8 + 3 = 7$ Thus, the vertex is $(2, 7)$.

2. Finding the Y-Intercept:

The y-intercept occurs when $x = 0$. Substituting $x = 0$ into the equation: $f(0) = -(0)^2 + 4(0) + 3 = 3$ Thus, the y-intercept is $(0, 3)$.

Summary:

• Vertex: $(2, 7)$
• Y-Intercept: $(0, 3)$

Would you like details on how to graph it, or do you have further questions?

Related Questions:

1. How do you graph quadratic functions step by step?
2. What is the importance of the axis of symmetry in a parabola?
3. How do you calculate the roots of a quadratic equation using factoring or the quadratic formula?
4. What is the significance of the sign of $a$ (whether the parabola opens upwards or downwards)?
5. How does the vertex form of a quadratic function compare to the standard form?

Tip:

The vertex form of a quadratic function is useful for quickly identifying the vertex without needing calculations. You can rewrite the standard form to the vertex form by completing the square!