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Complete the square to determine the vertex and axis of symmetry of y=−x2+8x+7.(4 points)

make it like a kid wrote it and write at least 3-5 sentences. Thank you.

3 answers

To complete the square for the equation \( y = -x^2 + 8x + 7 \), we first need to make the \( x \) terms nice. We can start by rewriting it a bit:

  1. Factor out the negative sign from the \( x^2 \) and \( x \) terms: \[ y = -(x^2 - 8x) + 7 \]

  2. Now, we want to make a perfect square inside the parentheses. To do this, we take half of the number in front of \( x \), which is \( -8 \). Half of \( 8 \) is \( 4 \), and then we square it, which gives us \( 16 \). We have to add and subtract \( 16 \) inside the parentheses: \[ y = -(x^2 - 8x + 16 - 16) + 7 \]

  3. Now we can rewrite it as: \[ y = -((x - 4)^2 - 16) + 7 \]

  4. Distributing the negative sign gives us: \[ y = -(x - 4)^2 + 16 + 7 \] which simplifies to: \[ y = -(x - 4)^2 + 23 \]

Now, we see that the vertex of the parabola is at the point \( (4, 23) \). The axis of symmetry is the vertical line that goes through the vertex, which is \( x = 4 \). So, the highest point of the parabola is at \( (4, 23) \) and that’s where it turns around! It makes a nice “U” shape, but upside down because of the negative sign!

make it simpler please

To find the vertex and axis of symmetry for the equation \( y = -x^2 + 8x + 7 \), we can complete the square. First, we group the \( x \) terms together:

  1. Write it as \( y = -(x^2 - 8x) + 7 \).
  2. To make a perfect square, we take half of \( 8 \) (which is \( 4 \)) and then square it (so we get \( 16 \)).
  3. We add and subtract \( 16 \) inside the parentheses: \[ y = -((x^2 - 8x + 16) - 16) + 7 \]
  4. This simplifies to: \[ y = -(x - 4)^2 + 23 \]

Now we can see that the vertex is at \( (4, 23) \). This means the highest point of the graph is at \( 4 \) on the \( x \)-axis and \( 23 \) on the \( y \)-axis. The axis of symmetry is the line \( x = 4 \), which goes straight up and down through the vertex. So, the graph is nice and symmetrical around this line!