Question

equation
$y_1 = x^2$
$y_2 = 3x^2$
$y_3 = 7x^2$
changes to parent graph

1. conjecture: multiplying the parent graph by a number whose absolute value is greater than one causes the parent graph to ______

equation
$y_1 = x^2$
$y_2 = ½x^2$
$y_3 = ¼x^2$
changes to parent graph

1. conjecture: multiplying the parent graph by a number whose absolute value is between zero and one causes the parent graph to ______

based on your conjectures above, sketch the graphs without using your graphing calculator.

1. $y = (x + 3)^2 - 4$
2. $y = -x^2 + 5$

this work is licensed under a creative commons attribution – noncommercial – sharealike 4.0 international license
© 2023 georgia department of education
all rights reserved
last updated july 2024
page 4 of 6

Explanation:

Step1: Analyze vertical scaling (|a|>1)

For \(y_1=x^2\), \(y_2=3x^2\), \(y_3=7x^2\): as \(|a|\) increases beyond 1, \(y\)-values grow faster, so the graph narrows.

Step2: Analyze vertical scaling (0<|a|<1)

For \(y_1=x^2\), \(y_2=\frac{1}{2}x^2\), \(y_3=\frac{1}{4}x^2\): as \(|a|\) is between 0 and 1, \(y\)-values grow slower, so the graph widens.

Step3: Sketch \(y=(x+3)^2-4\)

Parent \(y=x^2\) shifts left 3 (from \(x+3\)) and down 4 (from \(-4\)); vertex at \((-3,-4)\), opens up.

Step4: Sketch \(y=-x^2+5\)

Parent \(y=x^2\) reflects over x-axis (from \(-x^2\)) and shifts up 5 (from \(+5\)); vertex at \((0,5)\), opens down.

Answer:

1. narrow vertically
2. widen vertically
3. Parabola with vertex (-3,-4), opening upward
4. Parabola with vertex (0,5), opening downward

(Note: For sketching, plot the vertex first then a few points (e.g., for 8: when x=-2, y=-3; x=-4, y=-3; for 9: when x=1, y=4; x=-1, y=4) to draw the curve.)