Question

graph the following quadratic functions using your knowledge of transformations.
plot at least 5 points on each graph

1. $f(x) = 3x^2$

1. $f(x) = (x - 2)^2$

1. $f(x) = -x^2 + 3$

1. $f(x) = (x - 4)^2 + 1$

1. $f(x) = -(x + 3)^2 - 2$

1. $f(x) = \frac{1}{2}(x - 2)^2 + 1$

Explanation:

Step1: Analyze \( f(x) = 3x^2 \)

The parent function is \( y = x^2 \), which has vertex at \( (0,0) \), and points like \( (-2,4) \), \( (-1,1) \), \( (1,1) \), \( (2,4) \). For \( f(x) = 3x^2 \), the coefficient \( 3 \) vertically stretches the graph by a factor of \( 3 \). So new points: when \( x = -2 \), \( f(-2) = 3\times(-2)^2 = 12 \); \( x = -1 \), \( f(-1) = 3\times(-1)^2 = 3 \); \( x = 0 \), \( f(0) = 0 \); \( x = 1 \), \( f(1) = 3 \); \( x = 2 \), \( f(2) = 12 \). Plot these points \( (-2,12) \), \( (-1,3) \), \( (0,0) \), \( (1,3) \), \( (2,12) \) and draw the parabola opening upwards with vertex at origin, narrower than \( y = x^2 \).

Step2: Analyze \( f(x) = (x - 2)^2 \)

Parent function \( y = x^2 \) (vertex \( (0,0) \)). The transformation is a horizontal shift \( 2 \) units to the right (since \( x - 2 \) in the square). So vertex is \( (2,0) \). Points: \( x = 0 \), \( f(0) = (0 - 2)^2 = 4 \); \( x = 1 \), \( f(1) = (1 - 2)^2 = 1 \); \( x = 2 \), \( f(2) = 0 \); \( x = 3 \), \( f(3) = 1 \); \( x = 4 \), \( f(4) = 4 \). Plot \( (0,4) \), \( (1,1) \), \( (2,0) \), \( (3,1) \), \( (4,4) \), parabola opening upwards with vertex at \( (2,0) \).

Step3: Analyze \( f(x) = -x^2 + 3 \)

Parent \( y = x^2 \) (vertex \( (0,0) \)). The \( - \) reflects over the \( x \)-axis, and \( +3 \) shifts up \( 3 \) units. Vertex is \( (0,3) \). Points: \( x = -2 \), \( f(-2) = -(-2)^2 + 3 = -1 \); \( x = -1 \), \( f(-1) = -(-1)^2 + 3 = 2 \); \( x = 0 \), \( f(0) = 3 \); \( x = 1 \), \( f(1) = 2 \); \( x = 2 \), \( f(2) = -1 \). Plot \( (-2,-1) \), \( (-1,2) \), \( (0,3) \), \( (1,2) \), \( (2,-1) \), parabola opening downwards with vertex at \( (0,3) \).

Step4: Analyze \( f(x) = (x - 4)^2 + 1 \)

Parent \( y = x^2 \). Horizontal shift \( 4 \) units right (from \( x - 4 \)) and vertical shift \( 1 \) unit up. Vertex \( (4,1) \). Points: \( x = 2 \), \( f(2) = (2 - 4)^2 + 1 = 5 \); \( x = 3 \), \( f(3) = (3 - 4)^2 + 1 = 2 \); \( x = 4 \), \( f(4) = 1 \); \( x = 5 \), \( f(5) = 2 \); \( x = 6 \), \( f(6) = 5 \). Plot \( (2,5) \), \( (3,2) \), \( (4,1) \), \( (5,2) \), \( (6,5) \), parabola opening upwards with vertex at \( (4,1) \).

Step5: Analyze \( f(x) = -(x + 3)^2 - 2 \)

Parent \( y = x^2 \). Horizontal shift \( 3 \) units left ( \( x + 3 = x - (-3) \) ), reflection over \( x \)-axis ( \( - \) ), vertical shift \( 2 \) units down. Vertex \( (-3, -2) \). Points: \( x = -5 \), \( f(-5) = -(-5 + 3)^2 - 2 = -6 \); \( x = -4 \), \( f(-4) = -(-4 + 3)^2 - 2 = -3 \); \( x = -3 \), \( f(-3) = -2 \); \( x = -2 \), \( f(-2) = -3 \); \( x = -1 \), \( f(-1) = -6 \). Plot \( (-5,-6) \), \( (-4,-3) \), \( (-3,-2) \), \( (-2,-3) \), \( (-1,-6) \), parabola opening downwards with vertex at \( (-3, -2) \).

Step6: Analyze \( f(x) = \frac{1}{2}(x - 2)^2 + 1 \)

Parent \( y = x^2 \). Horizontal shift \( 2 \) units right, vertical stretch by factor \( \frac{1}{2} \) (compression), vertical shift \( 1 \) unit up. Vertex \( (2,1) \). Points: \( x = 0 \), \( f(0) = \frac{1}{2}(0 - 2)^2 + 1 = 3 \); \( x = 1 \), \( f(1) = \frac{1}{2}(1 - 2)^2 + 1 = 1.5 \); \( x = 2 \), \( f(2) = 1 \); \( x = 3 \), \( f(3) = 1.5 \); \( x = 4 \), \( f(4) = 3 \). Plot \( (0,3) \), \( (1,1.5) \), \( (2,1) \), \( (3,1.5) \), \( (4,3) \), parabola opening upwards, wider than \( y = x^2 \), vertex at \( (2,1) \).

Answer:

For each function, the graphs are drawn by plotting the calculated points (as per each step) and sketching the parabola with the correct vertex, direction, and shape based on transformations. (Note: Since the question asks to graph, the final answer is the graphical representation as per the steps above, with points plotted and parabolas drawn.)