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Graphing quadratics review

The graph of a quadratic function is a parabola, which is a "u"-shaped curve. In this article, we review how to graph quadratic functions.
The graph of a quadratic function is a parabola, which is a "u"-shaped curve:
A coordinate plane. The x- and y-axes both scale by one. The graph is the function x squared. The function is a parabola that opens up. The function decreases through negative two, four and negative one, one. The vertex of the function is plotted at zero, zero, then the function increases through one, one and two, four.
In this article, we review how to graph quadratic functions.
Looking for an introduction to parabolas? Check out this video.

Example 1: Vertex form

Graph the equation.
y, equals, minus, 2, left parenthesis, x, plus, 5, right parenthesis, squared, plus, 4

This equation is in vertex form.
y, equals, start color #e07d10, a, end color #e07d10, left parenthesis, x, minus, start color #11accd, h, end color #11accd, right parenthesis, squared, plus, start color #1fab54, k, end color #1fab54
This form reveals the vertex, left parenthesis, start color #11accd, h, end color #11accd, comma, start color #1fab54, k, end color #1fab54, right parenthesis, which in our case is left parenthesis, minus, 5, comma, 4, right parenthesis.
It also reveals whether the parabola opens up or down. Since start color #e07d10, a, end color #e07d10, equals, minus, 2, the parabola opens downward.
This is enough to start sketching the graph.
A coordinate plane. The x- and y-axes both scale by one. The graph is the function negative two times the sum of x plus five squared plus four. The function is a parabola that opens down. The vertex of the function is plotted at the point negative three, four and there are small lines leaving toward the rest of the function.
Incomplete sketch of y=-2(x+5)^2+4
To finish our graph, we need to find another point on the curve.
Let's plug x, equals, minus, 4 into the equation.
y=2(4+5)2+4=2(1)2+4=2+4=2\begin{aligned} y&=-2(-4+5)^2+4\\\\ &=-2(1)^2+4\\\\ &=-2+4\\\\ &=2 \end{aligned}
Therefore, another point on the parabola is left parenthesis, minus, 4, comma, 2, right parenthesis.
A coordinate plane. The x- and y-axes both scale by one. The graph is the function negative two times the sum of x plus five squared plus four. The function is a parabola that opens down. The vertex of the function is plotted at the point negative three, four. Another point is plotted at negative four, two.
Final graph of y=-2(x+5)^2+4
Want another example? Check out this video.

Example: Non-vertex form

Graph the function.
g, left parenthesis, x, right parenthesis, equals, x, squared, minus, x, minus, 6

First, let's find the zeros of the function—that is, let's figure out where this graph y, equals, g, left parenthesis, x, right parenthesis intersects the x-axis.
g(x)=x2x60=x2x60=(x3)(x+2)\begin{aligned} g(x)&=x^2-x-6 \\\\ 0&=x^2-x-6 \\\\ 0&=(x-3)(x+2) \end{aligned}
So our solutions are x, equals, 3 and x, equals, minus, 2, which means the points left parenthesis, minus, 2, comma, 0, right parenthesis and left parenthesis, 3, comma, 0, right parenthesis are where the parabola intersects the x-axis.
A coordinate plane. The x- and y-axes both scale by one. The points negative two, zero and three, zero are plotted.
To draw the rest of the parabola, it would help to find the vertex.
Parabolas are symmetric, so we can find the x-coordinate of the vertex by averaging the x-intercepts.
A coordinate plane. The x- and y-axes both scale by one. The points negative two, zero and three, zero are plotted. A point in plotted in the middle of these point at zero point five, zero.
The average of -2 and 3 is 0.5, which is the x-coordinate of our vertex.
With the x-coordinate figured out, we can solve for y by substituting into our original equation.
g(0.5)=(0.5)2(0.5)6=0.250.56=6.25\begin{aligned} g(\blueD{0.5})&=(\blueD{0.5})^2-(\blueD{0.5})-6 \\\\ &=0.25-0.5-6 \\\\ &=-6.25 \end{aligned}
Our vertex is at left parenthesis, 0, point, 5, comma, minus, 6, point, 25, right parenthesis, and our final graph looks like this:
A coordinate plane. The x- and y-axes both scale by one. The graph is the function x squared minus x minus six. The function is a parabola that opens up. The vertex of the function is plotted at the point zero point five, negative six point two-five. The x-intercepts are also plotted at negative two, zero and three, zero.
Graph of y=x^2-x-6
Want another example? Check out this video.

Practice

Problem 1
Graph the equation.
y, equals, 2, left parenthesis, x, plus, 1, right parenthesis, left parenthesis, x, minus, 1, right parenthesis

Want more practice graphing quadratics? Check out these exercises:

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