Let’s address the questions based on the "Vertex Form" of a quadratic equation, which typically takes the form:

$y = a(x - h)^2 + k$

Here:

• $(h, k)$ is the vertex of the parabola.
• $a$ controls the direction and width of the parabola.

Question 2: What information does a quadratic expression in the vertex form reveal? How does it show that information?

The vertex form directly reveals the vertex of the parabola, which is the point $(h, k)$. This is where the parabola reaches its maximum or minimum value (depending on the sign of $a$). The equation shows this information by having $(h, k)$ in the form $y = a(x - h)^2 + k$:

• $h$ is the x-coordinate of the vertex, which shows where the axis of symmetry is.
• $k$ is the y-coordinate of the vertex, indicating the highest or lowest point of the parabola.
• The value of $a$ determines the shape and orientation of the parabola:
  • If $a > 0$, the parabola opens upward (minimum at the vertex).
  • If $a < 0$, it opens downward (maximum at the vertex).
  • A larger value of $|a|$ makes the parabola narrower, while a smaller $|a|$ makes it wider.

Question 3: What doesn't it tell us?

While the vertex form gives us the vertex, it does not directly tell us:

• The x-intercepts (unless the equation is factored or solved).
• The exact location of the y-intercept, unless $x = 0$ is substituted into the equation to find it.
• General information about the symmetry of the parabola aside from the axis passing through $h$.

Question 4: Why do you think this form is used?

The vertex form is used because:

• It makes it easy to identify the vertex, which is crucial for graphing and analyzing the quadratic function.
• It helps in determining the direction and width of the parabola at a glance.
• It simplifies finding the maximum or minimum value of the quadratic function, which is useful in optimization problems.
• The form can also be used to transform the quadratic graph, such as shifting or reflecting it, based on the values of $h$, $k$, and $a$.

Would you like more details on any of these answers or further clarifications? Here are some related questions:

1. How do you find the x-intercepts of a quadratic equation given in vertex form?
2. What are the steps to convert standard form to vertex form?
3. How can the value of $a$ influence real-life applications of quadratic functions?
4. How is the vertex form applied in physics or engineering?
5. What is the relationship between the vertex form and completing the square?

Tip: The vertex form is especially helpful when solving word problems where the vertex (maximum or minimum) is the main point of interest, such as in projectile motion scenarios.