Determining Quadratic Function for a Graph

In summary, the conversation is about determining the quadratic function for a given graph, which has X-intercepts at (-3,0) and (5,0), a Y-intercept at (0,-30), and a local minimum/vertex at (1,-32). The person is seeking help with finding the correct equation for the parabola, and the expert suggests using the formula $y(x)=a(x-x_0)^2+y_0$ with the given points to solve for the value of $a$.
  • #1
Sharpy1
5
0
Hmm I'm having issues with an optional problem for some review for a quiz later tonight.

Determine the Quadratic Function for the Graph
the points labeled are X intercepts =(-3,0) (5,0); Y-Intercept = (0,-30), Local Min/Vertex = (1,-32)
View attachment 2420

It's the parabola looking problem at the bottom right, sorry I wasn't able to snap a good picture of it.

This is part of some optional review and I have a few hours before the quiz so any help would be appreciated. I'm honestly not even sure where to being other than putting "-32" at the end of the equation as the interpretation of the minimum point.
 

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  • #2
A parabola with vertex at $(x_0,y_0)$ has equation $y(x)=a(x-x_0)^2+y_0$ for some $a$. So, here the formula is $y(x)=a(x-1)^2-32$. Find $a$ from the fact that $y(5)=0$ or from $y(0)=-30$.
 

FAQ: Determining Quadratic Function for a Graph

What is a quadratic function?

A quadratic function is a type of polynomial function that can be written in the form of f(x) = ax^2 + bx + c, where a, b, and c are constants and x is the independent variable. It is a curve that forms a parabola and is commonly used to model real-world phenomena.

How do you determine a quadratic function from a graph?

To determine a quadratic function from a graph, you need to identify the key characteristics of the parabola, such as its vertex, axis of symmetry, and direction of opening. From there, you can use the general form of a quadratic function and plug in the values to find the specific equation.

What is the vertex of a parabola?

The vertex of a parabola is the point where the curve changes direction, either from increasing to decreasing or vice versa. It is also the highest or lowest point on the curve, depending on the direction of opening. In the general form of a quadratic function, the vertex can be found by using the formula x = -b/2a.

What is the axis of symmetry?

The axis of symmetry is an imaginary line that divides a parabola into two symmetrical halves. It passes through the vertex and is always perpendicular to the directrix, which is a line parallel to the axis of symmetry and located at a distance equal to the focal length from the vertex.

How do you determine the direction of opening for a parabola?

The direction of opening for a parabola is determined by the coefficient a in the general form of a quadratic function. If a is positive, the parabola opens upwards, and if a is negative, the parabola opens downwards. This can also be visualized by looking at the shape of the parabola on the graph.

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