Find the vertex of the graph of each quadratic function....

Find the vertex of the graph of each quadratic function. Determine
whether the graph opens upward or downward, find any intercepts, and sketch the graph. See Examples 1 through 4 .
$$
f(x)=x^{2}+4 x-5
$$

Key Concepts

Intercepts

Intercepts are the points where the graph intersects the coordinate axes. The y-intercept is obtained by evaluating the function at x = 0, while the x-intercepts (or roots) are found by solving the quadratic equation ax² + bx + c = 0, using methods such as factoring, completing the square, or the quadratic formula.

Direction of Opening

The direction in which a parabola opens is determined by the sign of the leading coefficient, a. If a is positive, the parabola opens upward, indicating that the vertex is the minimum point; if a is negative, it opens downward, making the vertex the maximum point.

Graphing a Parabola

Graphing a parabola involves plotting the vertex, intercepts, and additional points located symmetrically about the axis of symmetry. This process provides a clear visual understanding of the quadratic function and its key features.

Quadratic Function

A quadratic function is a polynomial of degree two, typically written in the form ax² + bx + c. Its graph is a parabola, and the coefficients determine its shape, position, and direction.

Vertex of a Parabola

The vertex is the point on the parabola that represents either the maximum or minimum value of the function. It can be found using the formula (-b/(2a), f(-b/(2a))), and it also serves as the axis of symmetry for the graph.

Transcript

00:01 Okay, right.

00:02 So in this question, you're given a quadratic function and you're asking to find the vertex of the quadratic function, number one.

00:12 Number two, you're asking to determine whether the graph opens upward or downward.

00:18 Number three, you're asking to find the intercepts.

00:22 And lastly, you're asking to sketch the graph.

00:26 Okay, so to start with finding the vertex of the quadratic function, we need to need.

00:31 To write this function in the completed square form.

00:39 Okay, so i'm going to write x squared here plus 4x, and i'm going to leave a space between the second and the third term.

00:49 Okay, and then i'm going to find the number that should be added to the second term to complete the square.

00:57 Okay, so what we do usually is half of the second term, so half of four.

01:03 And square that.

01:05 So half of four is two and when you square that you have four.

01:10 And so you realize also that four is not in the original equation.

01:14 So to even things out, i'll take away four.

01:18 Okay.

01:20 So to complete the square here, i'll have to write x plus two squared minus nine.

01:31 Okay.

01:32 So this has been written in the complete it's square form and so you can now see that the vertex the quadratic function will occur at the ordered pair negative to negative 9 okay so number one number two that was number one number two we need to find the we need to determine whether the function opens upward or downward.

02:07 Okay, so to do that we need to look at the leading coefficient at the quadratic function.

02:14 The leading coefficient a is one and that is clearly positive.

02:21 And when that happens, you can see that the the parabola here happens upward...

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