To match the graphs labeled J, K, and L with the correct quadratic equations in vertex form, we need to analyze the graphs in terms of their vertex positions and the direction of their parabolas (whether they open upwards or downwards). Vertex form of a quadratic equation is given by:

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

Where:

• $(h, k)$ is the vertex of the parabola.
• $a$ determines whether the parabola opens upwards ($a > 0$) or downwards ($a < 0$) and how "wide" or "narrow" it is.

Step-by-step analysis:

1. Graph J:

   • The parabola opens upwards.
   • The vertex appears to be at (0, 0).
   • This means $h = 0$, $k = 0$, and the equation should have a positive coefficient of $a$ since it opens upwards.

2. Graph K:

   • The parabola opens downwards.
   • The vertex appears to be at (2, 4).
   • This means $h = 2$, $k = 4$, and $a$ will be negative because the parabola opens downwards.

3. Graph L:

   • The parabola opens upwards.
   • The vertex is located at (-1, 1).
   • Therefore, $h = -1$, $k = 1$, and $a$ is positive.

Matching equations:

Now, let's look at the given options in vertex form:

• For Graph J (vertex at (0, 0), opens upwards): $y = x^2 \quad \text{(since the vertex is at (0,0) and opens upwards)}$

• For Graph K (vertex at (2, 4), opens downwards): $y = -2(x - 2)^2 + 4$

• For Graph L (vertex at (-1, 1), opens upwards): $y = (x + 1)^2 + 1$

Final matches:

• J: $y = x^2$
• K: $y = -2(x - 2)^2 + 4$
• L: $y = (x + 1)^2 + 1$

Let me know if you need further details or explanations!

Relative questions:

1. What is the significance of the vertex form of a quadratic equation?
2. How does changing the coefficient $a$ affect the graph of a quadratic function?
3. How can you determine whether a parabola opens upwards or downwards by looking at its equation?
4. What role do the parameters $h$ and $k$ play in the vertex form?
5. How can you graph a quadratic equation given in vertex form by hand?

Tip:

When analyzing quadratic graphs, always focus on identifying the vertex first. This simplifies matching the graph with its equation in vertex form.