Graph the quadratic function. Find the x

Graph the quadratic function. Find the $x$ - and $y$ -intercepts of each graph, if any exist. If it is given in general form, convert it into standard form; if it is given in standard form, convert it into general form. Find the domain and range of the function and list the intervals on which the function is increasing or decreasing. Identify the vertex and the axis of symmetry and determine whether the vertex yields a relative and absolute maximum or minimum. $f(x)=-3 x^{2}+4 x-7$

Key Concepts

Quadratic Functions

Quadratic functions are polynomials of degree 2 typically written in the form ax² + bx + c. They produce parabolic graphs that either open upward (a > 0) or downward (a < 0), and understanding their shape is fundamental for solving equations and optimization problems.

General Form and Standard Form

The general form of a quadratic function is ax² + bx + c, while the standard (or vertex) form is a(x - h)² + k, where (h, k) represents the vertex of the parabola. Converting between these forms helps reveal important features of the graph such as its vertex, direction, and width.

X- and Y-intercepts

The intercepts of a quadratic function are the points at which the graph crosses the coordinate axes. The x-intercepts are found by solving ax² + bx + c = 0, and the y-intercept is found by evaluating the function at x = 0. These points are essential for accurately sketching the graph of the function.

Domain and Range

The domain of a quadratic function is all real numbers since any x-value can be input into the function. The range, however, is determined by the vertex of the parabola; it includes all y-values above or below the vertex, depending on whether the parabola opens upward or downward.

Vertex and Axis of Symmetry

The vertex is the highest or lowest point on the graph of a quadratic function, depending on the orientation of the parabola. The axis of symmetry is a vertical line that passes directly through the vertex, dividing the parabola into two mirror-image halves and aiding in the analysis of the function’s behavior.

Intervals of Increase and Decrease

Quadratic functions have specific intervals where they are either increasing or decreasing. Generally, if the parabola opens upward, it decreases to the left of the vertex and increases to the right; if it opens downward, it increases to the left and decreases to the right. Identifying these intervals is useful in understanding the function's overall behavior.

Relative and Absolute Extrema

For quadratic functions, the vertex represents both the relative and absolute maximum or minimum value, depending on the direction in which the parabola opens. This concept is important for optimization, as it indicates the highest or lowest value the function can achieve.

Transcript

00:01 So now we have, again, a general form equation.

00:05 And unfortunately, this one, we're going to have to deal with fractions.

00:07 But we do what we have to do.

00:10 So let's put it into that standard form.

00:13 So we're going to need to factor negative three out, and that's going to become x squared minus four thirds.

00:19 And then we're going to leave a gap there, and we're going to put a minus seven.

00:23 And we know we need to make this be a perfect square inside here.

00:27 And so we need to take that negative four thirds and square it and take i should say take half of it and then square it so that's going to become negative two -thirds squared which will be four -nights so that it looks like we want to add four -nines again half of this which is two -thirds negative two -thirds and then squared so four nines but we really just added negative three times that and negative 3 times 4 9s is negative 4 thirds so we need to add 4 thirds right here and then i want this to be in the form of negative 21 over 3 there we go so now we have completed our square and we can factor this down and then we can combine these two together and that gives us negative 17 over 3 so we can see now where the vertex is, the vertex is at two thirds and negative 17 thirds.

01:38 We know that it goes downward, so we know that there's an absolute max.

01:45 And we could quick sketch a graph of this, but let's just do one other.

01:49 Well, we can even give, let's find the x and the y intercepts right now, x intercept and the y intercept.

01:56 And let's do the algebra for that.

01:57 I'll do that in red.

01:59 And so we're going to set this equal to zero.

02:02 And so we're going to end up having negative.

02:04 3 is times this x minus 2 thirds quantity squared and then move this 17 thirds to the other side oh and i see something's going to happen notice that if i multiply both sides by negative 1 3rd that i would have this quantity squared has to equal a negative value there will be no none there will not be an x intercept because this graph is going to have a vertex here and it's going to open downward.

02:37 So it's not going to have an x intercept, but it will have a y intercept.

02:41 And that's going to occur when i plug zero into my function.

02:45 If i plug zero in, i'm going to end up getting negative seven...

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