Translating Parabolas §5.3

By the end of today, you should be able to… Use the vertex form of a quadratic function to graph a parabola. Convert from Standard form to vertex form.

Investigation Vertex Form y = ax2 +bx+ c y = a(x-h)2 + k Each function in the first column is written in standard form. Each function has been rewritten in the second column in vertex form. Copy and complete the table. See if you can determine a relationship between h and . Standard Form y = ax2 +bx+ c -b 2a y Vertex Form y = a(x-h)2 + k h k y = x2 – 4x + 4 y = (x-2)2 y = x2 + 6x + 8 y = (x +3)2 - 1 y = -3x2 – 12x - 8 y = -3(x+2)2 + 4 The x-coordinate of the vertex of the standard form of the equation is the same as the value of h of the vertex form of the equation. The y-coordinate of the vertex of the standard form of the equation is the same as the value of k of the vertex form of the equation.

You can do the same with quadratic functions! Using Vertex Form In Chapter 2, you learned how to graph absolute value functions as translations of their parent graphs. You can do the same with quadratic functions! To translate the graph of a quadratic function, you can use the vertex form of a quadratic function: y = a(x - h)2 + k

Graph of a Quadratic Function in Vertex Form The graph of y = a(x - h)2 + k is the graph of y = ax2 translated h units horizontally and k units vertically.

Graph of a Quadratic Function in Vertex Form y = a(x - h)2 + k When h is positive the graph shifts right** When h is negative the graph shifts left** When k is positive the graph shifts up When k is negative the graph shifts down. The vertex is (h, k) The axis of symmetry is the line x = h. Graph of a Quadratic Function in Vertex Form

Example 1: Using Vertex Form to Graph a Parabola Graph the function as a translation of its parent function. Step 1) Graph the parent function. Step 2) Stretch or compress. Step 3) Move right or left. Step 4) Move up or down.

Example 2: Using Vertex Form to Graph a Parabola Graph the function as a translation of its parent function. Step 1) Graph the parent function. Step 2) Stretch or compress. Step 3) Move right or left. Step 4) Move up or down.

Example 3: Writing the Equation of a parabola. Write the equation of the parabola in vertex form. Step 1) Use vertex form: y = a(x-h)2 + k Step 2) Substitute the values of h and k. Step 3) Substitute another point. Step 4) Substitute a, h and k into the original equation.

Example 4: Writing the Equation of a parabola. Write the equation of the parabola in vertex form. Step 1) Use vertex form: y = a(x-h)2 + k Step 2) Substitute the values of h and k. Step 3) Substitute another point. Step 4) Substitute a, h and k into the original equation. You try this one!

Example 5: Converting to vertex form. Write the equation in vertex form. Step 1) Find the x-coordinate of the vertex. Step 2) Find the y-coordinate of the vertex. Step 3) Rewrite in vertex form. y = a(x-h)2 + k You try this one!

Example 6: Converting to vertex form. Write the equation in vertex form. Step 1) Find the x-coordinate of the vertex. Step 2) Find the y-coordinate of the vertex. Step 3) Rewrite in vertex form. y = a(x-h)2 + k You try this one!

Practice! Identify the vertex and the y-intercept of the graph of the function. Write each function in vertex form. (4, -25), -41 Y=(x-2)2+2 Y=6x2-10

Homework p.244 (14 – 34 even, 42)