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Algebra 1 9-12 / Week 11 / Loft Day
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Week 11 · Quadratic Functions and Their Graphs

Loft Day

Loft Day: sketch the arc
// The rocket's path is a parabola
⏱ about 20 min

Loft Day: Sketch the Arc

Saturday the graph paper covers the kitchen table and the family picks cards.

"Each card is a rule," Comet says. "Sketch its parabola before Nova projects the real one."

The first card reads y = (x - 2)² - 1. Someone sketches a U with its bottom at (2, -1).

"How did you know the vertex?" Wren asks. "It is right there in the rule," comes the answer.

Nova projects the true curve over the sketch. They match. "Would you like a harder one?"

The next card reads y = -x² + 4x. "Complete the square first," Comet whispers. "Then sketch."

"Vertex (2, 4), opens down, crosses at 0 and 4," says a voice. Nova's curve agrees.

  1. A quadratic function graphs as a parabola with a vertex, an axis of symmetry and intercepts.
  2. The vertex is a maximum when a is negative and a minimum when a is positive.
  3. Graph by hand from a table or from vertex form. Complete the square to find the vertex.
  4. Standard, vertex and factored forms each reveal a different feature of the same function.
  5. f(x) + k, k f(x), f(x + k) and f(kx) shift, stretch, flip and squeeze the graph.
◇ FAMILY SKETCH CARDS
Write six rules on cards: three in vertex form and three in standard form.
Deal a card. Everyone sketches the parabola, marking the vertex and the way it opens.
Then make a quick table of three points and check the sketch. For standard form, complete the square first.
For a twist, give a card like f(x) + 2 or f(x - 3) and have everyone slide the last sketch.
For a grown-up
This is a paper activity, no launch needed. If the family launches the rocket, a grown-up is present, everyone stands back and the rocket points away from people.
A sketch that is off is a good moment: check one point by substitution and see where the curve really goes.
Every rule on the cards is made up for practice, not a measurement of a real throw.
LOFT DAY CHECK
  • Read the question.
  • Tap your answer.
The first card is y = (x - 2)² - 1, which equals x² - 4x + 3. What is the vertex?
The second card is y = -x² + 4x. What is the vertex?
Where does y = -x² + 4x cross the x-axis? Solve -x² + 4x = 0.
For the card y = -x² + 4x, what is the maximum value of y? Type the number.
WHY THIS EXERCISEThe vertex of a parabola that opens down is its maximum. Symmetry finds it fast.
Draw the two Loft Day cards on one grid. Label each vertex and say which opens up and which opens down.
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