Use the graph to find the equation for the quadratic function. Write the function in standard fo

Use the graph to find the equation for the quadratic...

Key Concepts

Standard Form

The standard form of a quadratic function is given by f(x) = ax² + bx + c. This form is particularly useful because it clearly shows the coefficients, which affect the shape and position of the parabola. The constant term c represents the y-intercept, while the coefficients a and b influence the curvature and the location of the vertex of the parabola.

Graph Interpretation

Graph interpretation involves analyzing the key features of the parabola provided in the graph, such as the vertex, axis of symmetry, intercepts, and other identifiable points. These features allow us to form equations or systems of equations by substituting corresponding x and y values into the quadratic function, essential for determining the unknown coefficients.

Quadratic Function

A quadratic function is a second-degree polynomial function typically expressed as f(x) = ax² + bx + c. Its graph is a parabola, which opens upward if the coefficient a is positive and downward if a is negative. This function has important properties such as a vertex, a line of symmetry, and x- and y-intercepts that are key in understanding its behavior.

Determining Coefficients

Determining the coefficients of a quadratic function from its graph is done by using the coordinates of specific points on the graph. Once key points—typically including the vertex and one or more other points—are identified, their x and y values are substituted into the standard form equation. This process results in a system of equations that can be solved to find the values of a, b, and c, which fully define the quadratic function.

Transcript

00:01 So this equation, or a problem here, we're going to write an equation in the standard form.

00:06 Now, when we look at these, the first thing we want to do is identify our x intercepts.

00:10 You see, i have one and x is equal to negative three.

00:13 And we also have one at x as equal to positive two.

00:16 So what we're doing here is kind of unfactoring.

00:20 So in the last steps to solve them by factoring, we had the answers, and that came for when things were equal to zero.

00:26 So the first thing we want to do is set each of this equal to zero.

00:28 So i'm going to add 3 to this side here to get x plus 3 is equal to 0.

00:33 I'm going to subtract 2 from both sides here to get x minus 2 is equal to 0.

00:38 From there, i know that i have now factors of x plus 3 and x minus 2.

00:44 So these were the final steps to factoring, which is why we did it this way.

00:48 So now we need to find out, was there a number that was originally factored out here in order for us to get the equation? so the way that we figured that out is by identifying a point on the graph.

00:58 So when i look at this, it looks as if we have a y intercept of 0 comma negative 6.

01:04 So i'm going to plug x in for 0 and for x and for x and negative 6 in for y and solve.

01:10 And that will get me the value of a.

01:13 So now i substitute it in...

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