Question

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" in quadratic function \( y = a(x - h)^2 + k \)

• For a quadratic function in vertex form \( y = a(x - h)^2 + k \), the coefficient \( a \) affects the vertical stretch/compression and reflection.
• When the slider for \( a \) moves to the right (assuming \( a>0 \) and increasing), the value of \( a \) increases. If \( a>0 \), as \( |a| \) increases, the parabola vertically stretches (becomes narrower). If \( a \) was negative and moving right towards zero or positive, the parabola may reflect (if crossing zero) and then stretch.
• When the slider for \( a \) moves to the left, \( |a| \) either decreases (if \( a>0 \)) or becomes more negative (if \( a<0 \)). If \( a>0 \), decreasing \( |a| \) causes vertical compression (wider parabola). If \( a<0 \), more negative \( a \) means vertical stretch and reflection (opening down, narrower).
• When \( 0 < a < 1 \), the parabola is vertically compressed (wider than the parent function \( y = x^2 \)) because \( |a| < 1 \) causes a vertical compression.
• When \( a < 0 \), the parabola opens downward (reflection over the x - axis) and its width depends on \( |a| \) (stretch if \( |a|>1 \), compress if \( 0 < |a| < 1 \)).
• Conclusion about \( a \): The value of \( a \) determines the vertical stretch ( \( |a|>1 \) ), compression ( \( 0<|a|<1 \) ) of the parabola and its direction (opens up if \( a>0 \), down if \( a<0 \) ).

Step2: Analyze "h" in quadratic function \( y = a(x - h)^2 + k \)

• The vertex form \( y = a(x - h)^2 + k \) has its vertex at \( (h,k) \).
• When moving the slider for \( h \) to the right, the value of \( h \) increases. The graph of the parabola shifts horizontally to the right (since the vertex \( (h,k) \) moves right, and the parabola is symmetric about \( x = h \)).
• When moving the slider for \( h \) to the left, the value of \( h \) decreases, so the graph shifts horizontally to the left.
• A change in \( h \) results in a horizontal shift of the function. The direction of the shift is opposite to the sign in the equation: \( y = a(x - h)^2 + k \) shifts \( h \) units to the right if \( h>0 \), left if \( h<0 \). So moving the slider right (increasing \( h \)) shifts the graph right, moving left (decreasing \( h \)) shifts left.

Answer:

For "a" - related questions:

• When \( a \) slider moves right ( \( a \) increases, \( a>0 \) scenario): Parabola vertically stretches (narrower, if \( a>0 \)) or reflects/ stretches (if \( a \) crosses zero from negative to positive).
• When \( a \) slider moves left ( \( a \) decreases or becomes more negative): Parabola vertically compresses (wider, if \( a>0 \)) or stretches/ reflects (if \( a<0 \), more negative means narrower downward opening).
• When \( 0 < a < 1 \): Parabola is vertically compressed (wider than \( y = x^2 \)).
• When \( a < 0 \): Parabola opens downward (reflected over x - axis) with width based on \( |a| \).
• Conclusion on \( a \): \( a \) controls vertical stretch/compression ( \( |a| \) magnitude) and direction ( \( a>0 \) up, \( a<0 \) down).

For "h" - related questions:

• Moving \( h \) slider right: Graph shifts horizontally to the right (relative to parent graph \( y = x^2 \)).
• Moving \( h \) slider left: Graph shifts horizontally to the left.
• Change in \( h \): Horizontal shift (opposite to sign in \( (x - h) \), right for \( h>0 \) increase, left for \( h<0 \) decrease).