Answer:
- y = (x - 2)(x - 4)
- y = x² - 6x + 8
Step-by-step explanation:
To find the equation of the quadratic function given a diagram, note key features of the graph. These are marked on the photo attached.
- The roots or zeroes are: (2, 0) and (4, 0)
- The vertex is: (3, –1)
- The y-intercept is: (0, 8)
- If needed, we can also use other coordinate points on the function.
Quadratic equations
Quadratic equations that represent quadratic functions can be written in a variety of forms:
- Factored form, which highlights the roots: [tex]y = a(x - p)(x - q)[/tex]
- Vertex form, which highlights the vertex: [tex]y = a(x - h)^{2} + k[/tex]
- Standard form, which highlights the y-intercept: [tex]y = ax^{2} + bx + c[/tex]
Depending on the information we have, we can choose between the forms. Standard form is the hardest to find directly, but many final answers need to be in standard form.
We can convert a different form into standard form if needed. Let's start with factored form.
Solve using factored form
Factored form is y = a(x - p)(x - q), where the roots are:
- (x, p), which we will substitute with (2, 0);
- and (x, q), which we will substitute with (4, 0).
In our equation, we are missing the variable 'a'.
Start with the general equation for factored form.
y = a(x - p)(x - q)
Substitute the roots. Keep in mind that they will be the opposite negative/positive when in the equation.
y = a(x - 2)(x - 4)
Substitute 'x' and 'y' with any point as (x, y). Let's use the y-intercept (0, 8).
8 = a(0 - 2)(0 - 4)
Simplify within the brackets. Isolate the 'a' variable.
8 = a(–2)(–4)
Multiply.
8 = 8a
Divide both sides by 8. Keep the variable on the left.
a = 1
Rewrite in factored form using p = 2, q = 4, and a = 1.
y = a(x - p)(x - q) General equation for factored form.
y = 1(x - p)(x - q) Substitute.
However, we do not need to write the 1.
y = (x - p)(x - q) Final answer in factored form.
Expand into standard form
To find the standard form of the equation, expand the equation in factored form.
y = (x - 2)(x - 4) Start with factored form.
y = x² - 2x - 4x + 8 Use FOIL.
y = x² - 6x + 8 Final answer in standard form.
Therefore, the equation of the quadratic function is y = x² - 6x + 8.
Learn more about expanding factored form into standard form here:
https://brainly.com/question/31730312
