Standard form of a quadratic function
Convert quadratic functions to standard form ax²+bx+c in Grade 10 algebra, identify coefficients, and use them to find vertex, axis of symmetry, and direction of opening.
Key Concepts
A quadratic function is a function that can be written in the form $f(x) = ax^2 + bx + c$, where $a \neq 0$ and $a$, $b$, and $c$ are real numbers. This is also known as the standard form.
To write $y + 3x = 8 2x^2$ in standard form, rearrange the terms: $y = 2x^2 3x + 8$. To convert $f(x) = 3(x 2)^2 + 5$ to standard form, expand $(x 2)^2$, then distribute and combine terms: $f(x) = 3(x^2 4x + 4) + 5 = 3x^2 12x + 17$.
This is the official 'dress code' for quadratic functions. Every term lines up neatly by its power of x, from $x^2$ down to the constant. Arranging it this way makes it much easier to analyze the function's graph and behavior, like finding the y intercept, which is just 'c'! It turns messy equations into something we can work with.
Common Questions
What is the standard form of a quadratic function?
f(x) = ax²+bx+c where a≠0. The coefficient a determines direction (a>0 opens up, a<0 opens down) and width of the parabola.
How do you find the vertex from standard form ax²+bx+c?
The x-coordinate of the vertex is x = -b/(2a). Substitute this into the function to find the y-coordinate. The vertex is (-b/2a, f(-b/2a)).
How does the value of c relate to the y-intercept?
c is the y-intercept of the parabola, since f(0) = a(0)²+b(0)+c = c. It is where the parabola crosses the y-axis.