Using the Vertex Form
Using vertex form is a Grade 7 math skill from Yoshiwara Intermediate Algebra that teaches students to work with quadratic equations written as y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. This form makes it easy to identify the vertex and graph the parabola quickly.
Key Concepts
Property To find a parabola's equation from its vertex $(x v, y v)$ and another point $(x, y)$: 1. Substitute the vertex into the vertex form: $y = a(x x v)^2 + y v$. 2. Substitute the coordinates of the other point for $x$ and $y$. 3. Solve for the value of $a$. 4. Write the final equation with the value of $a$.
Examples A parabola has a vertex at $(3, 4)$ and passes through $(5, 12)$. Start with $y = a(x 3)^2 + 4$. Substitute the point: $12 = a(5 3)^2 + 4$, which gives $8 = 4a$, so $a=2$. The equation is $y = 2(x 3)^2 + 4$.
A ball's path has a vertex at $(8, 12)$ and starts at $(0, 4)$. Using $y = a(x 8)^2 + 12$, we plug in $(0,4)$: $4 = a(0 8)^2 + 12$. This gives $ 8 = 64a$, so $a = \frac{1}{8}$. The equation is $y = \frac{1}{8}(x 8)^2 + 12$.
Common Questions
What is vertex form of a quadratic?
Vertex form is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola and a determines the direction and width of the opening.
How do you find the vertex from vertex form?
In y = a(x - h)^2 + k, the vertex is directly given as (h, k). For example, y = 2(x - 3)^2 + 5 has vertex (3, 5).
How do you convert standard form to vertex form?
Complete the square on the standard form ax^2 + bx + c to rewrite it as a(x - h)^2 + k, identifying h = -b/(2a) and k = f(h).
What does the value of a tell you in vertex form?
If a > 0, the parabola opens upward (minimum at vertex). If a < 0, it opens downward (maximum at vertex). Larger |a| means a narrower parabola.