Question

relative to the \parent graph,\ what happens to your graph when you move the slider to the left?
based on your answers above, a change in the \k\ value results in what kind of shift to the function?
based on your manipulation of the sliders, how would you find the minimum or maximum? given the value of a, h, and k, would you be able to determine the coordinates of the minimum or maximum? if so, how?
quadratic functions
perform the following actions and answer accordingly based on what you observe.
move the slider for \a\.
what happens to the graph when the slider for a moves to the right?
what happens to the graph when the slider for \a\ moves to the left?
what happens to the graph when a is less than 1, but greater than 0?
what happens to the graph when a is less than 0?
based on the information from your answers above, what could you conclude about the value \a\?
move the slider for \h\.
relative to the \parent graph\ what happens to your graph when you move the slider to the right?
relative to the \parent graph,\ what happens to your graph when you move the slider to the left?
based on your answers above, a change in the \h\ value results in what kind of shift to the function?

Explanation:

Step1: Analyze "a" slider (vertical stretch/compression, reflection)

• When \( a > 0 \) and slider moves right ( \( a \) increases): The parabola becomes narrower (vertical stretch).
• When slider moves left ( \( a \) decreases towards 0 or negative): If \( a \) is positive, narrower to wider; if \( a \) becomes negative, parabola opens downward (reflection).
• For \( 0 < a < 1 \): Parabola widens (vertical compression).
• For \( a < 0 \): Parabola opens downward (reflection over x - axis), and width depends on \( |a| \).
• Conclusion on \( a \): \( |a| \) controls width (larger \( |a| \) = narrower, smaller \( |a| \) = wider); sign of \( a \) controls direction (positive = up, negative = down).

Step2: Analyze "h" slider (horizontal shift)

• Moving \( h \) slider right: Graph shifts right relative to parent \( y = ax^2 \).
• Moving \( h \) slider left: Graph shifts left relative to parent.
• Conclusion on \( h \): Change in \( h \) causes horizontal shift (right/left).

Step3: Analyze "k" (implied, vertical shift) & min/max

• For min/max: Vertex of \( y = a(x - h)^2 + k \) is \( (h, k) \). If \( a > 0 \), vertex is minimum; if \( a < 0 \), vertex is maximum. Coordinates of min/max are \( (h, k) \).
• For \( k \) shift: Change in \( k \) causes vertical shift (up/down).

Answer:

For "a" slider:

• Right: Narrower (vertical stretch, \( a > 0 \) increasing).
• Left: Wider (if \( a \) approaches 0) or opens downward (if \( a \) becomes negative).
• \( 0 < a < 1 \): Wider (vertical compression).
• \( a < 0 \): Opens downward (reflection), width by \( |a| \).
• Conclusion: \( |a| \) = width, sign of \( a \) = direction.

For "h" slider:

• Right: Graph shifts right.
• Left: Graph shifts left.
• Shift type: Horizontal shift (right/left).

For min/max:

• Find via vertex \( (h, k) \). If \( a > 0 \), \( (h, k) \) is min; if \( a < 0 \), \( (h, k) \) is max. Yes, coordinates are \( (h, k) \).

For "k" shift (implied):

• Change in \( k \) = vertical shift (up/down).