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Interpreting Quadratic Functions in Standard Form

Interpreting Quadratic Functions in Standard Form 1.11 - Solution

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We want to solve the given quadratic equation by graphing. To do so, we will graph the quadratic function represented by the left-hand side of the above equation. To draw the graph, we must start by identifying the values of and We can see that and Now, we will follow four steps to graph the function.

  1. Find the axis of symmetry.
  2. Calculate the vertex.
  3. Identify the intercept and its reflection across the axis of symmetry.
  4. Connect the points with a parabola.

Finding the Axis of Symmetry

The axis of symmetry is a vertical line with equation Since we already know the values of and we can substitute them into the formula.
The axis of symmetry of the parabola is the vertical line with equation

Calculating the Vertex

To calculate the vertex, we need to think of as a function of We can write the expression for the vertex by stating the and coordinates in terms of and Note that the formula for the coordinate is the same as the formula for the axis of symmetry, which is Thus, the coordinate of the vertex is also To find the coordinate, we need to substitute for in our function.
Simplify right-hand side
We found the coordinate, and now we know that the vertex is

Identifying the intercept and its Reflection

The intercept of the graph of a quadratic function written in standard form is given by the value of Thus, the point where our graph intercepts the axis is Let's plot this point and its reflection across the axis of symmetry.

Connecting the Points

We can now draw the graph of the function. Since which is positive, the parabola will open upward. Let's connect the three points with a smooth curve.

The intercepts of the graph are the solutions to the given equation. However, this graph has no intercepts. Therefore, this equation has no real solutions.