7.1. Graphing Quadratics Standard Form College Algebra 7.1. Graphing Quadratics Standard Form

Do Now: Solve each of the following problems 8+3 𝑥+2 2 =44 − 𝑥 2 =4𝑥 8+3 𝑥+2 2 =44 − 𝑥 2 =4𝑥 5 𝑥 2 =35 2 𝑥 2 +7𝑥+1=5 4 𝑥 2 −2𝑥−5=0

Do Now: Complete do now handed to you when you entered class

Homework Questions? Comments? Confusions? Concerns? ASK ASK ASK ASK ASK!

Goal: We currently know how to factor quadratics or things that look like 𝑎 𝑥 2 +𝑏𝑥+𝑐, but… what do they look like on a graph?

We know… Linear Equations 𝑦=𝑚𝑥+𝑏

We know… Exponential Equations

Now… Quadratic Equations The graph of 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 will always look like a U shape

Parabola: This U shape, is called a parabola

Parts of a Parabola

Parts of a Parabola Vertex: The highest or lowest point on a graph

Parts of a Parabola Axis of Symmetry: The vertical “mirror” of the graph that goes through the vertex. Everything on the left side of this line is mirrored on the right side.

Parts of a Parabola Y Intercept: Where the graph crosses the 𝑦−𝑎𝑥𝑖𝑠. There is always one and only one Y intercept for a quadratic equation.

Parts of a Parabola X Intercept: Where the graph crosses the 𝑥- axis. These are the solutions of the quadratic equation: If the equation has two solutions, it intersects the 𝑥−𝑎𝑥𝑖𝑠 twice. If the equation has one solution, it intersects the 𝑥−𝑎𝑥𝑖𝑠 once. If the equation has no solutions, it never intersects the 𝑥−𝑎𝑥𝑖𝑠

Visually: X intercepts Two Intercepts:

Visually: One Intercept One X Intercept:

Visually: X Intercepts No X Intercepts:

Realize: Parabolas can be open “up” (Happy Face) or “down” (Frowny Face)

Example One: Graph 𝑓 𝑥 = 𝑥 2 −4𝑥+3 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

How to Graph Parabolas: ** Our equation must be in the form 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 Step 1: We determine the direction of opening by checking if 𝑎 is positive or negative.

How to Graph Parabolas: Step Two: We find the vertex by… Find the value of − 𝑏 2𝑎 . This is the x coordinate of the vertex Plugging in this value for x in the equation yields the y coordinate of the vertex The axis of symmetry is 𝑥=− 𝑏 2𝑎 We plot the point we found and the axis of symmetry line

How to Graph Parabolas: Step Three: We find the y-intercept by plugging in zero for x. Plot our value on the y axis. If the point does not fit in your picture, do not graph it If the point does fit, graph the point and its mirror image on the other side of your axis of symmetry.

How to Graph Parabolas: Step Four: Find the x intercepts by solving the quadratic equation. Make 𝑦=0 and you can… Take the square root if 𝑏=0 Factor the GCF if 𝑐=0 Use PS2 if all three terms are present Quadratic Formula is all else fails

How to Graph Parabolas: Step Five: Draw your parabola. You must have at least three points plotted. Make sure that it is a U shape and not a V shape or straight line.

Example One: Graph 𝑓 𝑥 = 𝑥 2 −4𝑥+3 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

Example Two: Graph 𝑓 𝑥 = −𝑥 2 −8𝑥−12 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

Example Three: Graph 𝑓 𝑥 = 𝑥 2 −5𝑥 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

Do Now: Handed to you when you entered class

Example Four: Graph 𝑓 𝑥 = 𝑥 2 −4 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

You Try #1: Graph 𝑓 𝑥 = 𝑥 2 −8𝑥+15 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

You Try #2: Graph 𝑓 𝑥 =− 𝑥 2 +9𝑥−20 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

Realize: So far everything we have graphed has had two x intercepts! BUT there could be parabola’s with ONE x intercept!

Example Five: Graph 𝑓 𝑥 =−4 𝑥 2 +12𝑥−9 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

Example Six: Graph 𝑓 𝑥 =4 𝑥 2 −4𝑥+1 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

Example Seven: Graph 𝑓 𝑥 =− 𝑥 2 +6𝑥−9 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

Realize: To get another point, choose an x value close to the vertex and plug it in to get the y value! (x,y) gets you a point!

Example Eight: Graph 𝑓 𝑥 =9 𝑥 2 +36𝑥+36 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

You Try #1: Graph 𝑓 𝑥 = 𝑥 2 −2𝑥+1 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

You Try #2: Graph 𝑓 𝑥 =−4 𝑥 2 +24𝑥−36 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

Do Now: Take out homework packet and complete #6

So…. We know how to graph things that have one intercept or TWO x intercepts…. What about NO x intercepts?

Example Ten: Graph 𝑓 𝑥 = 𝑥 2 −4𝑥+8 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

Example Eleven: Graph 𝑓 𝑥 =− 𝑥 2 +5𝑥−7 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

Example Twelve: Graph 𝑓 𝑥 = 1 2 𝑥 2 +2𝑥+9 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

Example Thirteen: Graph 𝑓 𝑥 =−2 𝑥 2 −1 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

You Try #1: Graph 𝑓 𝑥 =− 𝑥 2 +4𝑥−5 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

You Try #2: Graph 𝑓 𝑥 =− 𝑥 2 +4𝑥−9 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

You Try #3: Graph 𝑓 𝑥 = 𝑥 2 +8𝑥+15 Direction of Opening Vertex/Axis of Symmetry Y intercept X Intercepts Domain Range

Practice Problems Try some on your own/in your table groups As always don’t hesitate to ask questions if you are confused OR ask your tablemates– they are your greatest resource!