Question
Graph the quadratic equation y=4x^2+3x-8. What is the vertex of this quadratic equation?
Answers
Answered by
GPT-4o mini
To find the vertex of the quadratic equation \( y = 4x^2 + 3x - 8 \), we can use the vertex formula given by \( x = -\frac{b}{2a} \), where \( a \) and \( b \) are the coefficients from the standard form of the quadratic equation \( y = ax^2 + bx + c \).
From the equation \( y = 4x^2 + 3x - 8 \):
- \( a = 4 \)
- \( b = 3 \)
- \( c = -8 \)
Now, we can calculate the \( x \)-coordinate of the vertex:
\[
x = -\frac{b}{2a} = -\frac{3}{2 \times 4} = -\frac{3}{8}
\]
Next, we substitute \( x = -\frac{3}{8} \) back into the quadratic equation to find the \( y \)-coordinate of the vertex:
\[
y = 4\left(-\frac{3}{8}\right)^2 + 3\left(-\frac{3}{8}\right) - 8
\]
Calculating each term step-by-step:
1. \( \left(-\frac{3}{8}\right)^2 = \frac{9}{64} \)
2. \( 4 \times \frac{9}{64} = \frac{36}{64} = \frac{9}{16} \)
3. \( 3 \times -\frac{3}{8} = -\frac{9}{8} = -\frac{72}{64} \) (to get a common denominator)
Now, we can combine these results:
\[
y = \frac{9}{16} - \frac{72}{64} - 8
\]
To simplify this, convert -8 into fractional form with a denominator of 64:
\(-8 = -\frac{512}{64}\)
Now, we can put everything together:
\[
y = \frac{9}{16} - \frac{72}{64} - \frac{512}{64}
\]
Convert \(\frac{9}{16}\) to a common denominator of 64:
\(\frac{9}{16} = \frac{36}{64}\)
Now we have:
\[
y = \frac{36}{64} - \frac{72}{64} - \frac{512}{64} = \frac{36 - 72 - 512}{64} = \frac{-548}{64} = -\frac{137}{16}
\]
Thus, the vertex of the quadratic equation is:
\[
\left(-\frac{3}{8}, -\frac{137}{16}\right)
\]
### Summary
- The vertex of the quadratic equation \( y = 4x^2 + 3x - 8 \) is \( \left(-\frac{3}{8}, -\frac{137}{16}\right) \).
To graph the quadratic equation, you would plot the vertex and additional points, then draw the parabolic shape opening upwards since the coefficient \( a = 4 \) is positive.
From the equation \( y = 4x^2 + 3x - 8 \):
- \( a = 4 \)
- \( b = 3 \)
- \( c = -8 \)
Now, we can calculate the \( x \)-coordinate of the vertex:
\[
x = -\frac{b}{2a} = -\frac{3}{2 \times 4} = -\frac{3}{8}
\]
Next, we substitute \( x = -\frac{3}{8} \) back into the quadratic equation to find the \( y \)-coordinate of the vertex:
\[
y = 4\left(-\frac{3}{8}\right)^2 + 3\left(-\frac{3}{8}\right) - 8
\]
Calculating each term step-by-step:
1. \( \left(-\frac{3}{8}\right)^2 = \frac{9}{64} \)
2. \( 4 \times \frac{9}{64} = \frac{36}{64} = \frac{9}{16} \)
3. \( 3 \times -\frac{3}{8} = -\frac{9}{8} = -\frac{72}{64} \) (to get a common denominator)
Now, we can combine these results:
\[
y = \frac{9}{16} - \frac{72}{64} - 8
\]
To simplify this, convert -8 into fractional form with a denominator of 64:
\(-8 = -\frac{512}{64}\)
Now, we can put everything together:
\[
y = \frac{9}{16} - \frac{72}{64} - \frac{512}{64}
\]
Convert \(\frac{9}{16}\) to a common denominator of 64:
\(\frac{9}{16} = \frac{36}{64}\)
Now we have:
\[
y = \frac{36}{64} - \frac{72}{64} - \frac{512}{64} = \frac{36 - 72 - 512}{64} = \frac{-548}{64} = -\frac{137}{16}
\]
Thus, the vertex of the quadratic equation is:
\[
\left(-\frac{3}{8}, -\frac{137}{16}\right)
\]
### Summary
- The vertex of the quadratic equation \( y = 4x^2 + 3x - 8 \) is \( \left(-\frac{3}{8}, -\frac{137}{16}\right) \).
To graph the quadratic equation, you would plot the vertex and additional points, then draw the parabolic shape opening upwards since the coefficient \( a = 4 \) is positive.