Graph the equation. $y=\frac{2}{3} x^2+4 x+4$
Answer (2)
Find the vertex of the parabola using x = − 2 a b and substitute to find the y-coordinate.
Determine the x-intercepts by setting y = 0 and solving the quadratic equation using the quadratic formula.
Find the y-intercept by setting x = 0 .
Plot the vertex, x-intercepts, and y-intercept to sketch the parabola. The vertex is ( − 3 , − 2 ) , the x-intercepts are approximately − 1.268 and − 4.732 , and the y-intercept is ( 0 , 4 ) .
See graph.
Explanation
Analyze the problem We are given the equation y = 3 2 x 2 + 4 x + 4 and asked to graph it. This is a quadratic equation, which represents a parabola. To graph it, we'll find the vertex, x-intercepts, and y-intercept.
Find the vertex First, let's find the vertex of the parabola. The x-coordinate of the vertex is given by x = − 2 a b , where a = 3 2 and b = 4 . Thus, x = − 2 ( 3 2 ) 4 = − 3 4 4 = − 4 × 4 3 = − 3. Now, we can find the y-coordinate of the vertex by plugging x = − 3 into the equation: y = 3 2 ( − 3 ) 2 + 4 ( − 3 ) + 4 = 3 2 ( 9 ) − 12 + 4 = 6 − 12 + 4 = − 2. So, the vertex is at ( − 3 , − 2 ) .
Find the x-intercepts Next, let's find the x-intercepts by setting y = 0 : 3 2 x 2 + 4 x + 4 = 0 Multiplying by 3/2 to simplify: x 2 + 6 x + 6 = 0 Using the quadratic formula, x = 2 a − b ± b 2 − 4 a c , where a = 1 , b = 6 , and c = 6 : x = 2 ( 1 ) − 6 ± 6 2 − 4 ( 1 ) ( 6 ) = 2 − 6 ± 36 − 24 = 2 − 6 ± 12 = 2 − 6 ± 2 3 = − 3 ± 3 So the x-intercepts are approximately x = − 3 + 3 ≈ − 1.268 and x = − 3 − 3 ≈ − 4.732 .
Find the y-intercept Now, let's find the y-intercept by setting x = 0 : y = 3 2 ( 0 ) 2 + 4 ( 0 ) + 4 = 4 So, the y-intercept is at ( 0 , 4 ) .
Sketch the parabola Now we have the vertex ( − 3 , − 2 ) , the x-intercepts (approximately − 1.268 and − 4.732 ), and the y-intercept ( 0 , 4 ) . We can plot these points and sketch the parabola.
Final Answer The vertex of the parabola is ( − 3 , − 2 ) . The x-intercepts are approximately − 1.268 and − 4.732 . The y-intercept is ( 0 , 4 ) .
Examples
Understanding parabolas is crucial in various real-world applications, such as designing satellite dishes or understanding projectile motion. For instance, if you throw a ball, its path roughly follows a parabolic trajectory. By knowing the equation of this parabola, you can predict how far the ball will travel and where it will land. Similarly, satellite dishes are shaped like parabolas to focus incoming signals onto a single point, maximizing signal strength. The vertex and intercepts of the parabola help in optimizing the design for these applications.
The vertex of the parabola is at ( − 3 , − 2 ) , with x-intercepts approximately at − 1.268 and − 4.732 , and a y-intercept at ( 0 , 4 ) . These points help sketch the parabola represented by the equation y = 3 2 x 2 + 4 x + 4 . The graph opens upwards due to the positive coefficient of x 2 .
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